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Added electric field calculations and corrected a bug in the fringe
spacing function
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@@ -61,9 +61,15 @@
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<li>
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<a class="function" href="#resistivity">resistivity</a>
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</li>
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<li>
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<a class="function" href="#electric_field_strength_two_points">electric_field_strength_two_points</a>
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</li>
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<li>
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<a class="function" href="#fringe_spacing">fringe_spacing</a>
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</li>
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<li>
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<a class="function" href="#diffraction_grating_wavelength">diffraction_grating_wavelength</a>
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</li>
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</ul>
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@@ -85,88 +91,147 @@
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<label class="view-source-button" for="mod-physics-view-source"><span>View Source</span></label>
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<div class="pdoc-code codehilite"><pre><span></span><span id="L-1"><a href="#L-1"><span class="linenos"> 1</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">random</span>
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</span><span id="L-2"><a href="#L-2"><span class="linenos"> 2</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">math</span>
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</span><span id="L-3"><a href="#L-3"><span class="linenos"> 3</span></a>
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</span><span id="L-4"><a href="#L-4"><span class="linenos"> 4</span></a><span class="c1"># Generic</span>
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</span><span id="L-5"><a href="#L-5"><span class="linenos"> 5</span></a><span class="k">def</span><span class="w"> </span><span class="nf">kinetic_energy</span><span class="p">(</span><span class="n">max_mass</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">max_vel</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
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</span><span id="L-6"><a href="#L-6"><span class="linenos"> 6</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Kinetic Energy calculation using Ek = 0.5 * m * v^2</span>
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</span><span id="L-7"><a href="#L-7"><span class="linenos"> 7</span></a>
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</span><span id="L-8"><a href="#L-8"><span class="linenos"> 8</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
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</span><span id="L-9"><a href="#L-9"><span class="linenos"> 9</span></a><span class="sd"> | --- | --- |</span>
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</span><span id="L-10"><a href="#L-10"><span class="linenos">10</span></a><span class="sd"> | What is the kinetic energy of an object of mass $5 kg$ and velocity $10 m/s$ | $250 J$ |</span>
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</span><span id="L-11"><a href="#L-11"><span class="linenos">11</span></a><span class="sd"> """</span>
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</span><span id="L-12"><a href="#L-12"><span class="linenos">12</span></a> <span class="n">velocity</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_vel</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
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</span><span id="L-13"><a href="#L-13"><span class="linenos">13</span></a> <span class="n">mass</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_mass</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
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</span><span id="L-14"><a href="#L-14"><span class="linenos">14</span></a> <span class="n">kinetic_energy</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">mass</span> <span class="o">*</span> <span class="n">velocity</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
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</span><span id="L-15"><a href="#L-15"><span class="linenos">15</span></a>
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</span><span id="L-16"><a href="#L-16"><span class="linenos">16</span></a>
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</span><span id="L-17"><a href="#L-17"><span class="linenos">17</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"What is the kinetic energy of an object of mass $</span><span class="si">{</span><span class="n">mass</span><span class="si">}</span><span class="s2"> kg$ and velocity $</span><span class="si">{</span><span class="n">velocity</span><span class="si">}</span><span class="s2"> m/s$?"</span>
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</span><span id="L-18"><a href="#L-18"><span class="linenos">18</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s1">'$</span><span class="si">{</span><span class="n">kinetic_energy</span><span class="si">}</span><span class="s1"> J$'</span>
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</span><span id="L-19"><a href="#L-19"><span class="linenos">19</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
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</span><span id="L-20"><a href="#L-20"><span class="linenos">20</span></a>
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</span><span id="L-21"><a href="#L-21"><span class="linenos">21</span></a>
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</span><span id="L-22"><a href="#L-22"><span class="linenos">22</span></a><span class="c1"># Electricity</span>
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</span><span id="L-23"><a href="#L-23"><span class="linenos">23</span></a><span class="k">def</span><span class="w"> </span><span class="nf">potential_dividers</span><span class="p">(</span><span class="n">max_vin</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">max_resistance</span><span class="o">=</span><span class="mi">500</span><span class="p">):</span>
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</span><span id="L-24"><a href="#L-24"><span class="linenos">24</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Potential Divider question using Vout = (Vin * R2) / (R2 + R1)</span>
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</span><span id="L-25"><a href="#L-25"><span class="linenos">25</span></a>
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</span><span id="L-26"><a href="#L-26"><span class="linenos">26</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
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</span><span id="L-27"><a href="#L-27"><span class="linenos">27</span></a><span class="sd"> | --- | --- |</span>
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</span><span id="L-28"><a href="#L-28"><span class="linenos">28</span></a><span class="sd"> | In a Potential Divider, if resistors R1 and R2 have resistances of $100 \Omega$ and $50 \Omega$ respectively, and the cell has $12 V$ What is the output potential difference across R2? | $4 V$ |</span>
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</span><span id="L-29"><a href="#L-29"><span class="linenos">29</span></a><span class="sd"> """</span>
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</span><span id="L-30"><a href="#L-30"><span class="linenos">30</span></a><span class="w"> </span><span class="sd">'''</span>
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</span><span id="L-31"><a href="#L-31"><span class="linenos">31</span></a><span class="sd"> This is what a potential divider circuit looks like:</span>
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</span><span id="L-32"><a href="#L-32"><span class="linenos">32</span></a><span class="sd"> ------</span>
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</span><span id="L-33"><a href="#L-33"><span class="linenos">33</span></a><span class="sd"> | R1</span>
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</span><span id="L-34"><a href="#L-34"><span class="linenos">34</span></a><span class="sd"> Vi = |----o</span>
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</span><span id="L-35"><a href="#L-35"><span class="linenos">35</span></a><span class="sd"> | R2 Vout</span>
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</span><span id="L-36"><a href="#L-36"><span class="linenos">36</span></a><span class="sd"> |____|____o</span>
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</span><span id="L-37"><a href="#L-37"><span class="linenos">37</span></a><span class="sd"> '''</span>
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</span><span id="L-38"><a href="#L-38"><span class="linenos">38</span></a> <span class="n">vin</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_vin</span><span class="p">)</span> <span class="c1"># Voltage input of cell</span>
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</span><span id="L-39"><a href="#L-39"><span class="linenos">39</span></a> <span class="n">r1</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">)</span> <span class="c1"># Resistance of R1</span>
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</span><span id="L-40"><a href="#L-40"><span class="linenos">40</span></a> <span class="n">r2</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">)</span> <span class="c1"># Resistance of R2</span>
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</span><span id="L-41"><a href="#L-41"><span class="linenos">41</span></a> <span class="n">vout</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((</span><span class="n">vin</span> <span class="o">*</span> <span class="n">r2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">r1</span> <span class="o">+</span> <span class="n">r2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Voltage output across R2</span>
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</span><span id="L-42"><a href="#L-42"><span class="linenos">42</span></a>
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</span><span id="L-43"><a href="#L-43"><span class="linenos">43</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"In a Potential Divider, if resistors R1 and R2 have resistances of $</span><span class="si">{</span><span class="n">r1</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega$ and $</span><span class="si">{</span><span class="n">r2</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega$ respectively, and the cell has $</span><span class="si">{</span><span class="n">vin</span><span class="si">}</span><span class="s2"> V$ What is the output potential difference across R2?"</span>
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</span><span id="L-44"><a href="#L-44"><span class="linenos">44</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">vout</span><span class="si">}</span><span class="s2"> V$"</span>
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</span><span id="L-45"><a href="#L-45"><span class="linenos">45</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
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</span><span id="L-46"><a href="#L-46"><span class="linenos">46</span></a>
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</span><span id="L-47"><a href="#L-47"><span class="linenos">47</span></a><span class="k">def</span><span class="w"> </span><span class="nf">resistivity</span><span class="p">(</span><span class="n">max_diameter_mm</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">max_length_cm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_resistance</span><span class="o">=</span><span class="mf">0.1</span><span class="p">):</span>
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</span><span id="L-48"><a href="#L-48"><span class="linenos">48</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the Resistivity using the equation R = (pL)/A, where R = Resistance, L = length of wire, p = resistivity and A = cross sectional area of wire</span>
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</span><span id="L-49"><a href="#L-49"><span class="linenos">49</span></a>
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</span><span id="L-50"><a href="#L-50"><span class="linenos">50</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
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</span><span id="L-51"><a href="#L-51"><span class="linenos">51</span></a><span class="sd"> | --- | --- |</span>
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</span><span id="L-52"><a href="#L-52"><span class="linenos">52</span></a><span class="sd"> | A wire has resistance $30 m\Omega$ when it is $83.64 cm$ long with a diameter of $4.67 mm$. Calculate the resistivity of the wire | $6.14e-07 \Omega m$ |</span>
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</span><span id="L-53"><a href="#L-53"><span class="linenos">53</span></a><span class="sd"> """</span>
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</span><span id="L-54"><a href="#L-54"><span class="linenos">54</span></a> <span class="c1"># This question requires a lot of unit conversions and calculating the area of a circle from diameter</span>
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</span><span id="L-55"><a href="#L-55"><span class="linenos">55</span></a> <span class="n">diameter_mm</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_diameter_mm</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random diameter in mm</span>
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</span><span id="L-56"><a href="#L-56"><span class="linenos">56</span></a> <span class="n">cross_sectional_area</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="p">(</span><span class="n">diameter_mm</span> <span class="o">/</span> <span class="mi">2000</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># Calculate the cross sectional area using pi r²</span>
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</span><span id="L-57"><a href="#L-57"><span class="linenos">57</span></a> <span class="n">length_cm</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_length_cm</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random wire length in cm</span>
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</span><span id="L-58"><a href="#L-58"><span class="linenos">58</span></a> <span class="n">resistance</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random reistance in ohms</span>
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</span><span id="L-59"><a href="#L-59"><span class="linenos">59</span></a>
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</span><span id="L-60"><a href="#L-60"><span class="linenos">60</span></a> <span class="n">resistivity</span> <span class="o">=</span> <span class="p">(</span><span class="n">resistance</span> <span class="o">*</span> <span class="n">cross_sectional_area</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">length_cm</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
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</span><span id="L-61"><a href="#L-61"><span class="linenos">61</span></a>
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</span><span id="L-62"><a href="#L-62"><span class="linenos">62</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A wire has resistance $</span><span class="si">{</span><span class="n">resistance</span><span class="o">*</span><span class="mi">1000</span><span class="si">}</span><span class="s2"> m</span><span class="se">\\</span><span class="s2">Omega$ when it is $</span><span class="si">{</span><span class="n">length_cm</span><span class="si">}</span><span class="s2"> cm$ long with a diameter of $</span><span class="si">{</span><span class="n">diameter_mm</span><span class="si">}</span><span class="s2"> mm$. Calculate the resistivity of the wire"</span>
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</span><span id="L-63"><a href="#L-63"><span class="linenos">63</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">resistivity</span><span class="si">:</span><span class="s2">.2e</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega m$"</span>
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</span><span id="L-64"><a href="#L-64"><span class="linenos">64</span></a>
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</span><span id="L-65"><a href="#L-65"><span class="linenos">65</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
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</span><span id="L-66"><a href="#L-66"><span class="linenos">66</span></a>
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</span><span id="L-67"><a href="#L-67"><span class="linenos">67</span></a><span class="c1"># Waves</span>
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</span><span id="L-68"><a href="#L-68"><span class="linenos">68</span></a><span class="k">def</span><span class="w"> </span><span class="nf">fringe_spacing</span><span class="p">(</span><span class="n">max_screen_distance</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
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</span><span id="L-69"><a href="#L-69"><span class="linenos">69</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the fringe spacing in a double slit experiment with w=(λD)/s</span>
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</span><span id="L-70"><a href="#L-70"><span class="linenos">70</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
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</span><span id="L-71"><a href="#L-71"><span class="linenos">71</span></a><span class="sd"> | --- | --- |</span>
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</span><span id="L-72"><a href="#L-72"><span class="linenos">72</span></a><span class="sd"> | A laser with a wavelength of $450nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $12m$ from the slits and the slits are $0.30mm$ apart. Calculate the spacing between the bright fringes. | Using the equation $\\frac{{\\lambda D}}{{s}}$, we get a fringe spacing of $0.018m$ |</span>
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</span><span id="L-73"><a href="#L-73"><span class="linenos">73</span></a><span class="sd"> """</span>
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</span><span id="L-74"><a href="#L-74"><span class="linenos">74</span></a> <span class="n">wavelength_nm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">380</span><span class="p">,</span><span class="mi">750</span><span class="p">)</span> <span class="c1"># Random wavelength between violet and red (nm)</span>
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</span><span id="L-75"><a href="#L-75"><span class="linenos">75</span></a> <span class="n">screen_distance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_screen_distance</span><span class="p">)</span> <span class="c1"># Random distance between screen and slits (m)</span>
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</span><span id="L-76"><a href="#L-76"><span class="linenos">76</span></a> <span class="n">slit_spacing_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="p">)</span> <span class="c1"># Random slit spacing (mm)</span>
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</span><span id="L-77"><a href="#L-77"><span class="linenos">77</span></a>
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</span><span id="L-78"><a href="#L-78"><span class="linenos">78</span></a> <span class="n">fringe_spacing</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((((</span><span class="n">wavelength_nm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span> <span class="o">*</span> <span class="n">screen_distance</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">slit_spacing_mm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">3</span><span class="p">)),</span><span class="mi">5</span><span class="p">)</span>
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</span><span id="L-79"><a href="#L-79"><span class="linenos">79</span></a>
|
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</span><span id="L-80"><a href="#L-80"><span class="linenos">80</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser with a wavelength of $</span><span class="si">{</span><span class="n">wavelength_nm</span><span class="si">}</span><span class="s2">nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $</span><span class="si">{</span><span class="n">screen_distance</span><span class="si">}</span><span class="s2">m$ from the slits and the slits are $</span><span class="si">{</span><span class="n">slit_spacing_mm</span><span class="si">}</span><span class="s2">mm$ apart. Calculate the spacing between the bright fringes."</span>
|
||||
</span><span id="L-81"><a href="#L-81"><span class="linenos">81</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Using the equation $</span><span class="se">\\</span><span class="s2">frac</span><span class="se">{{\\</span><span class="s2">lambda D</span><span class="se">}}{{</span><span class="s2">s</span><span class="se">}}</span><span class="s2">$, we get a fringe spacing of $</span><span class="si">{</span><span class="n">fringe_spacing</span><span class="si">}</span><span class="s2">m$"</span>
|
||||
</span><span id="L-82"><a href="#L-82"><span class="linenos">82</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="L-1"><a href="#L-1"><span class="linenos"> 1</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">random</span>
|
||||
</span><span id="L-2"><a href="#L-2"><span class="linenos"> 2</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">math</span>
|
||||
</span><span id="L-3"><a href="#L-3"><span class="linenos"> 3</span></a>
|
||||
</span><span id="L-4"><a href="#L-4"><span class="linenos"> 4</span></a><span class="c1"># Mechanics</span>
|
||||
</span><span id="L-5"><a href="#L-5"><span class="linenos"> 5</span></a><span class="k">def</span><span class="w"> </span><span class="nf">kinetic_energy</span><span class="p">(</span><span class="n">max_mass</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">max_vel</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
|
||||
</span><span id="L-6"><a href="#L-6"><span class="linenos"> 6</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Kinetic Energy calculation using Ek = 0.5 * m * v^2</span>
|
||||
</span><span id="L-7"><a href="#L-7"><span class="linenos"> 7</span></a>
|
||||
</span><span id="L-8"><a href="#L-8"><span class="linenos"> 8</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-9"><a href="#L-9"><span class="linenos"> 9</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-10"><a href="#L-10"><span class="linenos"> 10</span></a><span class="sd"> | What is the kinetic energy of an object of mass $5 kg$ and velocity $10 m/s$ | $250 J$ |</span>
|
||||
</span><span id="L-11"><a href="#L-11"><span class="linenos"> 11</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-12"><a href="#L-12"><span class="linenos"> 12</span></a> <span class="n">velocity</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_vel</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
|
||||
</span><span id="L-13"><a href="#L-13"><span class="linenos"> 13</span></a> <span class="n">mass</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_mass</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
|
||||
</span><span id="L-14"><a href="#L-14"><span class="linenos"> 14</span></a> <span class="n">kinetic_energy</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">mass</span> <span class="o">*</span> <span class="n">velocity</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
|
||||
</span><span id="L-15"><a href="#L-15"><span class="linenos"> 15</span></a>
|
||||
</span><span id="L-16"><a href="#L-16"><span class="linenos"> 16</span></a>
|
||||
</span><span id="L-17"><a href="#L-17"><span class="linenos"> 17</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"What is the kinetic energy of an object of mass $</span><span class="si">{</span><span class="n">mass</span><span class="si">}</span><span class="s2"> kg$ and velocity $</span><span class="si">{</span><span class="n">velocity</span><span class="si">}</span><span class="s2"> m/s$?"</span>
|
||||
</span><span id="L-18"><a href="#L-18"><span class="linenos"> 18</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s1">'$</span><span class="si">{</span><span class="n">kinetic_energy</span><span class="si">}</span><span class="s1"> J$'</span>
|
||||
</span><span id="L-19"><a href="#L-19"><span class="linenos"> 19</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span><span id="L-20"><a href="#L-20"><span class="linenos"> 20</span></a>
|
||||
</span><span id="L-21"><a href="#L-21"><span class="linenos"> 21</span></a>
|
||||
</span><span id="L-22"><a href="#L-22"><span class="linenos"> 22</span></a><span class="c1"># Electricity & Electric Fields</span>
|
||||
</span><span id="L-23"><a href="#L-23"><span class="linenos"> 23</span></a><span class="k">def</span><span class="w"> </span><span class="nf">potential_dividers</span><span class="p">(</span><span class="n">max_vin</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">max_resistance</span><span class="o">=</span><span class="mi">500</span><span class="p">):</span>
|
||||
</span><span id="L-24"><a href="#L-24"><span class="linenos"> 24</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Potential Divider question using Vout = (Vin * R2) / (R2 + R1)</span>
|
||||
</span><span id="L-25"><a href="#L-25"><span class="linenos"> 25</span></a>
|
||||
</span><span id="L-26"><a href="#L-26"><span class="linenos"> 26</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-27"><a href="#L-27"><span class="linenos"> 27</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-28"><a href="#L-28"><span class="linenos"> 28</span></a><span class="sd"> | In a Potential Divider, if resistors R1 and R2 have resistances of $100 \Omega$ and $50 \Omega$ respectively, and the cell has $12 V$ What is the output potential difference across R2? | $4 V$ |</span>
|
||||
</span><span id="L-29"><a href="#L-29"><span class="linenos"> 29</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-30"><a href="#L-30"><span class="linenos"> 30</span></a><span class="w"> </span><span class="sd">'''</span>
|
||||
</span><span id="L-31"><a href="#L-31"><span class="linenos"> 31</span></a><span class="sd"> This is what a potential divider circuit looks like:</span>
|
||||
</span><span id="L-32"><a href="#L-32"><span class="linenos"> 32</span></a><span class="sd"> ------</span>
|
||||
</span><span id="L-33"><a href="#L-33"><span class="linenos"> 33</span></a><span class="sd"> | R1</span>
|
||||
</span><span id="L-34"><a href="#L-34"><span class="linenos"> 34</span></a><span class="sd"> Vi = |----o</span>
|
||||
</span><span id="L-35"><a href="#L-35"><span class="linenos"> 35</span></a><span class="sd"> | R2 Vout</span>
|
||||
</span><span id="L-36"><a href="#L-36"><span class="linenos"> 36</span></a><span class="sd"> |____|____o</span>
|
||||
</span><span id="L-37"><a href="#L-37"><span class="linenos"> 37</span></a><span class="sd"> '''</span>
|
||||
</span><span id="L-38"><a href="#L-38"><span class="linenos"> 38</span></a> <span class="n">vin</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_vin</span><span class="p">)</span> <span class="c1"># Voltage input of cell</span>
|
||||
</span><span id="L-39"><a href="#L-39"><span class="linenos"> 39</span></a> <span class="n">r1</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">)</span> <span class="c1"># Resistance of R1</span>
|
||||
</span><span id="L-40"><a href="#L-40"><span class="linenos"> 40</span></a> <span class="n">r2</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">)</span> <span class="c1"># Resistance of R2</span>
|
||||
</span><span id="L-41"><a href="#L-41"><span class="linenos"> 41</span></a> <span class="n">vout</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((</span><span class="n">vin</span> <span class="o">*</span> <span class="n">r2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">r1</span> <span class="o">+</span> <span class="n">r2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Voltage output across R2</span>
|
||||
</span><span id="L-42"><a href="#L-42"><span class="linenos"> 42</span></a>
|
||||
</span><span id="L-43"><a href="#L-43"><span class="linenos"> 43</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"In a Potential Divider, if resistors R1 and R2 have resistances of $</span><span class="si">{</span><span class="n">r1</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega$ and $</span><span class="si">{</span><span class="n">r2</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega$ respectively, and the cell has $</span><span class="si">{</span><span class="n">vin</span><span class="si">}</span><span class="s2"> V$ What is the output potential difference across R2?"</span>
|
||||
</span><span id="L-44"><a href="#L-44"><span class="linenos"> 44</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">vout</span><span class="si">}</span><span class="s2"> V$"</span>
|
||||
</span><span id="L-45"><a href="#L-45"><span class="linenos"> 45</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span><span id="L-46"><a href="#L-46"><span class="linenos"> 46</span></a>
|
||||
</span><span id="L-47"><a href="#L-47"><span class="linenos"> 47</span></a><span class="k">def</span><span class="w"> </span><span class="nf">resistivity</span><span class="p">(</span><span class="n">max_diameter_mm</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">max_length_cm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_resistance</span><span class="o">=</span><span class="mf">0.1</span><span class="p">):</span>
|
||||
</span><span id="L-48"><a href="#L-48"><span class="linenos"> 48</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the Resistivity using the equation R = (pL)/A, where R = Resistance, L = length of wire, p = resistivity and A = cross sectional area of wire</span>
|
||||
</span><span id="L-49"><a href="#L-49"><span class="linenos"> 49</span></a>
|
||||
</span><span id="L-50"><a href="#L-50"><span class="linenos"> 50</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-51"><a href="#L-51"><span class="linenos"> 51</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-52"><a href="#L-52"><span class="linenos"> 52</span></a><span class="sd"> | A wire has resistance $30 m\Omega$ when it is $83.64 cm$ long with a diameter of $4.67 mm$. Calculate the resistivity of the wire | $6.14e-07 \Omega m$ |</span>
|
||||
</span><span id="L-53"><a href="#L-53"><span class="linenos"> 53</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-54"><a href="#L-54"><span class="linenos"> 54</span></a> <span class="c1"># This question requires a lot of unit conversions and calculating the area of a circle from diameter</span>
|
||||
</span><span id="L-55"><a href="#L-55"><span class="linenos"> 55</span></a> <span class="n">diameter_mm</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_diameter_mm</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random diameter in mm</span>
|
||||
</span><span id="L-56"><a href="#L-56"><span class="linenos"> 56</span></a> <span class="n">cross_sectional_area</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="p">(</span><span class="n">diameter_mm</span> <span class="o">/</span> <span class="mi">2000</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># Calculate the cross sectional area using pi r²</span>
|
||||
</span><span id="L-57"><a href="#L-57"><span class="linenos"> 57</span></a> <span class="n">length_cm</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_length_cm</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random wire length in cm</span>
|
||||
</span><span id="L-58"><a href="#L-58"><span class="linenos"> 58</span></a> <span class="n">resistance</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_resistance</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Random reistance in ohms</span>
|
||||
</span><span id="L-59"><a href="#L-59"><span class="linenos"> 59</span></a>
|
||||
</span><span id="L-60"><a href="#L-60"><span class="linenos"> 60</span></a> <span class="n">resistivity</span> <span class="o">=</span> <span class="p">(</span><span class="n">resistance</span> <span class="o">*</span> <span class="n">cross_sectional_area</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">length_cm</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span>
|
||||
</span><span id="L-61"><a href="#L-61"><span class="linenos"> 61</span></a>
|
||||
</span><span id="L-62"><a href="#L-62"><span class="linenos"> 62</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A wire has resistance $</span><span class="si">{</span><span class="n">resistance</span><span class="o">*</span><span class="mi">1000</span><span class="si">}</span><span class="s2"> m</span><span class="se">\\</span><span class="s2">Omega$ when it is $</span><span class="si">{</span><span class="n">length_cm</span><span class="si">}</span><span class="s2"> cm$ long with a diameter of $</span><span class="si">{</span><span class="n">diameter_mm</span><span class="si">}</span><span class="s2"> mm$. Calculate the resistivity of the wire"</span>
|
||||
</span><span id="L-63"><a href="#L-63"><span class="linenos"> 63</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">resistivity</span><span class="si">:</span><span class="s2">.2e</span><span class="si">}</span><span class="s2"> </span><span class="se">\\</span><span class="s2">Omega m$"</span>
|
||||
</span><span id="L-64"><a href="#L-64"><span class="linenos"> 64</span></a>
|
||||
</span><span id="L-65"><a href="#L-65"><span class="linenos"> 65</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span><span id="L-66"><a href="#L-66"><span class="linenos"> 66</span></a>
|
||||
</span><span id="L-67"><a href="#L-67"><span class="linenos"> 67</span></a><span class="k">def</span><span class="w"> </span><span class="nf">electric_field_strength_two_points</span><span class="p">(</span><span class="n">max_seperation_cm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_charge_uC</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
|
||||
</span><span id="L-68"><a href="#L-68"><span class="linenos"> 68</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the total electric field strength at point P with given points A and B, using the equation kQ/r²</span>
|
||||
</span><span id="L-69"><a href="#L-69"><span class="linenos"> 69</span></a>
|
||||
</span><span id="L-70"><a href="#L-70"><span class="linenos"> 70</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-71"><a href="#L-71"><span class="linenos"> 71</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-72"><a href="#L-72"><span class="linenos"> 72</span></a><span class="sd"> | Charges A and B and point P are arranged like this: B <-- 7 cm --> P <-- 79 cm --> A, Where A and B have charges of -56 µC and -410 µC, What is the electric field strength at point P? | $-751417824 NC^{-1}$ (to the right) |</span>
|
||||
</span><span id="L-73"><a href="#L-73"><span class="linenos"> 73</span></a>
|
||||
</span><span id="L-74"><a href="#L-74"><span class="linenos"> 74</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-75"><a href="#L-75"><span class="linenos"> 75</span></a> <span class="n">a_charge</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_charge_uC</span><span class="p">,</span><span class="n">max_charge_uC</span><span class="p">)</span>
|
||||
</span><span id="L-76"><a href="#L-76"><span class="linenos"> 76</span></a> <span class="n">b_charge</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_charge_uC</span><span class="p">,</span><span class="n">max_charge_uC</span><span class="p">)</span>
|
||||
</span><span id="L-77"><a href="#L-77"><span class="linenos"> 77</span></a> <span class="n">arrangement</span> <span class="o">=</span> <span class="p">[[</span><span class="s1">'P'</span><span class="p">],[</span><span class="s1">'A'</span><span class="p">,</span><span class="n">a_charge</span><span class="p">],[</span><span class="s1">'B'</span><span class="p">,</span><span class="n">b_charge</span><span class="p">]]</span> <span class="c1"># Arrangement of charge A, B and the point of focus</span>
|
||||
</span><span id="L-78"><a href="#L-78"><span class="linenos"> 78</span></a> <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">arrangement</span><span class="p">)</span>
|
||||
</span><span id="L-79"><a href="#L-79"><span class="linenos"> 79</span></a> <span class="n">seperations</span> <span class="o">=</span> <span class="p">[</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">max_seperation_cm</span><span class="p">),</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">max_seperation_cm</span><span class="p">)]</span>
|
||||
</span><span id="L-80"><a href="#L-80"><span class="linenos"> 80</span></a> <span class="n">total_efs</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
</span><span id="L-81"><a href="#L-81"><span class="linenos"> 81</span></a> <span class="c1"># Work out how far A and B are from P (vector)</span>
|
||||
</span><span id="L-82"><a href="#L-82"><span class="linenos"> 82</span></a> <span class="k">if</span> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="L-83"><a href="#L-83"><span class="linenos"> 83</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
</span><span id="L-84"><a href="#L-84"><span class="linenos"> 84</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="L-85"><a href="#L-85"><span class="linenos"> 85</span></a> <span class="k">elif</span> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="L-86"><a href="#L-86"><span class="linenos"> 86</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
</span><span id="L-87"><a href="#L-87"><span class="linenos"> 87</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="L-88"><a href="#L-88"><span class="linenos"> 88</span></a> <span class="k">else</span><span class="p">:</span>
|
||||
</span><span id="L-89"><a href="#L-89"><span class="linenos"> 89</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
|
||||
</span><span id="L-90"><a href="#L-90"><span class="linenos"> 90</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="L-91"><a href="#L-91"><span class="linenos"> 91</span></a>
|
||||
</span><span id="L-92"><a href="#L-92"><span class="linenos"> 92</span></a> <span class="c1"># Work out the EFS at point P caused by A and B seperatley, then sum them together in `total_efs`</span>
|
||||
</span><span id="L-93"><a href="#L-93"><span class="linenos"> 93</span></a> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">arrangement</span><span class="p">:</span>
|
||||
</span><span id="L-94"><a href="#L-94"><span class="linenos"> 94</span></a> <span class="k">if</span> <span class="n">point</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="L-95"><a href="#L-95"><span class="linenos"> 95</span></a> <span class="k">continue</span>
|
||||
</span><span id="L-96"><a href="#L-96"><span class="linenos"> 96</span></a> <span class="k">else</span><span class="p">:</span>
|
||||
</span><span id="L-97"><a href="#L-97"><span class="linenos"> 97</span></a> <span class="n">efs</span> <span class="o">=</span> <span class="p">((</span><span class="mf">8.99</span><span class="o">*</span><span class="mi">10</span><span class="o">**</span><span class="mi">9</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">point</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="o">**-</span><span class="mi">6</span><span class="p">))</span><span class="o">/</span><span class="p">((</span><span class="n">point</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># efs = kQ/r²</span>
|
||||
</span><span id="L-98"><a href="#L-98"><span class="linenos"> 98</span></a> <span class="k">if</span> <span class="n">point</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span> <span class="n">efs</span> <span class="o">=</span> <span class="o">-</span><span class="n">efs</span>
|
||||
</span><span id="L-99"><a href="#L-99"><span class="linenos"> 99</span></a> <span class="n">point</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">efs</span><span class="p">)</span>
|
||||
</span><span id="L-100"><a href="#L-100"><span class="linenos">100</span></a> <span class="n">total_efs</span> <span class="o">+=</span> <span class="n">efs</span>
|
||||
</span><span id="L-101"><a href="#L-101"><span class="linenos">101</span></a>
|
||||
</span><span id="L-102"><a href="#L-102"><span class="linenos">102</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Charges A and B and point P are arranged like this:</span><span class="se">\n</span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> <-- $</span><span class="si">{</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">$ cm --> </span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> <-- $</span><span class="si">{</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">$ cm --> </span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="se">\n</span><span class="s2">Where A and B have charges of $</span><span class="si">{</span><span class="n">a_charge</span><span class="si">}</span><span class="s2">$ µC and $</span><span class="si">{</span><span class="n">b_charge</span><span class="si">}</span><span class="s2">$ µC</span><span class="se">\n</span><span class="s2">What is the electric field strength at point P?"</span>
|
||||
</span><span id="L-103"><a href="#L-103"><span class="linenos">103</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">total_efs</span><span class="p">)</span><span class="si">}</span><span class="s2"> NC^</span><span class="si">{</span><span class="o">-</span><span class="mi">1</span><span class="si">}</span><span class="s2">$ (to the right)"</span>
|
||||
</span><span id="L-104"><a href="#L-104"><span class="linenos">104</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span><span id="L-105"><a href="#L-105"><span class="linenos">105</span></a>
|
||||
</span><span id="L-106"><a href="#L-106"><span class="linenos">106</span></a>
|
||||
</span><span id="L-107"><a href="#L-107"><span class="linenos">107</span></a><span class="c1"># Waves</span>
|
||||
</span><span id="L-108"><a href="#L-108"><span class="linenos">108</span></a><span class="k">def</span><span class="w"> </span><span class="nf">fringe_spacing</span><span class="p">(</span><span class="n">max_screen_distance</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
|
||||
</span><span id="L-109"><a href="#L-109"><span class="linenos">109</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the fringe spacing in a double slit experiment with w=(λD)/s</span>
|
||||
</span><span id="L-110"><a href="#L-110"><span class="linenos">110</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-111"><a href="#L-111"><span class="linenos">111</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-112"><a href="#L-112"><span class="linenos">112</span></a><span class="sd"> | A laser with a wavelength of $450nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $12m$ from the slits and the slits are $0.30mm$ apart. Calculate the spacing between the bright fringes. | Using the equation $\\frac{{\\lambda D}}{{s}}$, we get a fringe spacing of $0.018m$ |</span>
|
||||
</span><span id="L-113"><a href="#L-113"><span class="linenos">113</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-114"><a href="#L-114"><span class="linenos">114</span></a> <span class="n">wavelength_nm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">380</span><span class="p">,</span><span class="mi">750</span><span class="p">)</span> <span class="c1"># Random wavelength between violet and red (nm)</span>
|
||||
</span><span id="L-115"><a href="#L-115"><span class="linenos">115</span></a> <span class="n">screen_distance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_screen_distance</span><span class="p">)</span> <span class="c1"># Random distance between screen and slits (m)</span>
|
||||
</span><span id="L-116"><a href="#L-116"><span class="linenos">116</span></a> <span class="n">slit_spacing_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="p">)</span> <span class="c1"># Random slit spacing (mm)</span>
|
||||
</span><span id="L-117"><a href="#L-117"><span class="linenos">117</span></a>
|
||||
</span><span id="L-118"><a href="#L-118"><span class="linenos">118</span></a> <span class="n">fringe_spacing</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((((</span><span class="n">wavelength_nm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span> <span class="o">*</span> <span class="n">screen_distance</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">slit_spacing_mm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">3</span><span class="p">)),</span><span class="mi">5</span><span class="p">)</span>
|
||||
</span><span id="L-119"><a href="#L-119"><span class="linenos">119</span></a>
|
||||
</span><span id="L-120"><a href="#L-120"><span class="linenos">120</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser with a wavelength of $</span><span class="si">{</span><span class="n">wavelength_nm</span><span class="si">}</span><span class="s2">nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $</span><span class="si">{</span><span class="n">screen_distance</span><span class="si">}</span><span class="s2">m$ from the slits and the slits are $</span><span class="si">{</span><span class="n">slit_spacing_mm</span><span class="si">}</span><span class="s2">mm$ apart. Calculate the spacing between the bright fringes."</span>
|
||||
</span><span id="L-121"><a href="#L-121"><span class="linenos">121</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Using the equation $</span><span class="se">\\</span><span class="s2">frac</span><span class="se">{{\\</span><span class="s2">lambda D</span><span class="se">}}{{</span><span class="s2">s</span><span class="se">}}</span><span class="s2">$, we get a fringe spacing of $</span><span class="si">{</span><span class="n">fringe_spacing</span><span class="si">}</span><span class="s2">m$"</span>
|
||||
</span><span id="L-122"><a href="#L-122"><span class="linenos">122</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span><span id="L-123"><a href="#L-123"><span class="linenos">123</span></a>
|
||||
</span><span id="L-124"><a href="#L-124"><span class="linenos">124</span></a><span class="k">def</span><span class="w"> </span><span class="nf">diffraction_grating_wavelength</span><span class="p">(</span><span class="n">min_slits_per_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_slits_per_mm</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_order_number</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
|
||||
</span><span id="L-125"><a href="#L-125"><span class="linenos">125</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the wavelength when given the number of slits per mm, order number and angle of order using the equation nλ = dsinθ</span>
|
||||
</span><span id="L-126"><a href="#L-126"><span class="linenos">126</span></a>
|
||||
</span><span id="L-127"><a href="#L-127"><span class="linenos">127</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="L-128"><a href="#L-128"><span class="linenos">128</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="L-129"><a href="#L-129"><span class="linenos">129</span></a><span class="sd"> | A laser is shone through a diffraction grating which has $293$ lines per mm, the fringe of order number $2$ is at an angle of $0.39$ rad. Calculate the wavelength of the light | $\lambda = 6.487856913364529e-07m = 649nm |</span>
|
||||
</span><span id="L-130"><a href="#L-130"><span class="linenos">130</span></a>
|
||||
</span><span id="L-131"><a href="#L-131"><span class="linenos">131</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-132"><a href="#L-132"><span class="linenos">132</span></a> <span class="n">slits_per_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">min_slits_per_mm</span><span class="p">,</span> <span class="n">max_slits_per_mm</span><span class="p">)</span>
|
||||
</span><span id="L-133"><a href="#L-133"><span class="linenos">133</span></a> <span class="n">slit_spacing</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">slits_per_mm</span> <span class="o">*</span> <span class="mi">1000</span><span class="p">)</span>
|
||||
</span><span id="L-134"><a href="#L-134"><span class="linenos">134</span></a> <span class="n">order_number</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_order_number</span><span class="p">)</span>
|
||||
</span><span id="L-135"><a href="#L-135"><span class="linenos">135</span></a> <span class="n">angle_of_order</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="mf">0.2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
|
||||
</span><span id="L-136"><a href="#L-136"><span class="linenos">136</span></a> <span class="n">wavelength</span> <span class="o">=</span> <span class="p">((</span><span class="n">slit_spacing</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle_of_order</span><span class="p">))</span> <span class="o">/</span> <span class="n">order_number</span><span class="p">)</span>
|
||||
</span><span id="L-137"><a href="#L-137"><span class="linenos">137</span></a>
|
||||
</span><span id="L-138"><a href="#L-138"><span class="linenos">138</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser is shone through a diffraction grating which has $</span><span class="si">{</span><span class="n">slits_per_mm</span><span class="si">}</span><span class="s2">$ lines per mm, the fringe of order number $</span><span class="si">{</span><span class="n">order_number</span><span class="si">}</span><span class="s2">$ is at an angle of $</span><span class="si">{</span><span class="n">angle_of_order</span><span class="si">}</span><span class="s2">$ rad. Calculate the wavelength of the light"</span>
|
||||
</span><span id="L-139"><a href="#L-139"><span class="linenos">139</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="se">\\</span><span class="s2">lambda = </span><span class="si">{</span><span class="n">wavelength</span><span class="si">}</span><span class="s2">m = </span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">wavelength</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span><span class="si">}</span><span class="s2">nm$"</span>
|
||||
</span><span id="L-140"><a href="#L-140"><span class="linenos">140</span></a>
|
||||
</span><span id="L-141"><a href="#L-141"><span class="linenos">141</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
@@ -329,6 +394,78 @@
|
||||
</div>
|
||||
|
||||
|
||||
</section>
|
||||
<section id="electric_field_strength_two_points">
|
||||
<input id="electric_field_strength_two_points-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
|
||||
<div class="attr function">
|
||||
|
||||
<span class="def">def</span>
|
||||
<span class="name">electric_field_strength_two_points</span><span class="signature pdoc-code condensed">(<span class="param"><span class="n">max_seperation_cm</span><span class="o">=</span><span class="mi">100</span>, </span><span class="param"><span class="n">max_charge_uC</span><span class="o">=</span><span class="mi">1000</span></span><span class="return-annotation">):</span></span>
|
||||
|
||||
<label class="view-source-button" for="electric_field_strength_two_points-view-source"><span>View Source</span></label>
|
||||
|
||||
</div>
|
||||
<a class="headerlink" href="#electric_field_strength_two_points"></a>
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="electric_field_strength_two_points-68"><a href="#electric_field_strength_two_points-68"><span class="linenos"> 68</span></a><span class="k">def</span><span class="w"> </span><span class="nf">electric_field_strength_two_points</span><span class="p">(</span><span class="n">max_seperation_cm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_charge_uC</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
|
||||
</span><span id="electric_field_strength_two_points-69"><a href="#electric_field_strength_two_points-69"><span class="linenos"> 69</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the total electric field strength at point P with given points A and B, using the equation kQ/r²</span>
|
||||
</span><span id="electric_field_strength_two_points-70"><a href="#electric_field_strength_two_points-70"><span class="linenos"> 70</span></a>
|
||||
</span><span id="electric_field_strength_two_points-71"><a href="#electric_field_strength_two_points-71"><span class="linenos"> 71</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="electric_field_strength_two_points-72"><a href="#electric_field_strength_two_points-72"><span class="linenos"> 72</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="electric_field_strength_two_points-73"><a href="#electric_field_strength_two_points-73"><span class="linenos"> 73</span></a><span class="sd"> | Charges A and B and point P are arranged like this: B <-- 7 cm --> P <-- 79 cm --> A, Where A and B have charges of -56 µC and -410 µC, What is the electric field strength at point P? | $-751417824 NC^{-1}$ (to the right) |</span>
|
||||
</span><span id="electric_field_strength_two_points-74"><a href="#electric_field_strength_two_points-74"><span class="linenos"> 74</span></a>
|
||||
</span><span id="electric_field_strength_two_points-75"><a href="#electric_field_strength_two_points-75"><span class="linenos"> 75</span></a><span class="sd"> """</span>
|
||||
</span><span id="electric_field_strength_two_points-76"><a href="#electric_field_strength_two_points-76"><span class="linenos"> 76</span></a> <span class="n">a_charge</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_charge_uC</span><span class="p">,</span><span class="n">max_charge_uC</span><span class="p">)</span>
|
||||
</span><span id="electric_field_strength_two_points-77"><a href="#electric_field_strength_two_points-77"><span class="linenos"> 77</span></a> <span class="n">b_charge</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_charge_uC</span><span class="p">,</span><span class="n">max_charge_uC</span><span class="p">)</span>
|
||||
</span><span id="electric_field_strength_two_points-78"><a href="#electric_field_strength_two_points-78"><span class="linenos"> 78</span></a> <span class="n">arrangement</span> <span class="o">=</span> <span class="p">[[</span><span class="s1">'P'</span><span class="p">],[</span><span class="s1">'A'</span><span class="p">,</span><span class="n">a_charge</span><span class="p">],[</span><span class="s1">'B'</span><span class="p">,</span><span class="n">b_charge</span><span class="p">]]</span> <span class="c1"># Arrangement of charge A, B and the point of focus</span>
|
||||
</span><span id="electric_field_strength_two_points-79"><a href="#electric_field_strength_two_points-79"><span class="linenos"> 79</span></a> <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">arrangement</span><span class="p">)</span>
|
||||
</span><span id="electric_field_strength_two_points-80"><a href="#electric_field_strength_two_points-80"><span class="linenos"> 80</span></a> <span class="n">seperations</span> <span class="o">=</span> <span class="p">[</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">max_seperation_cm</span><span class="p">),</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">max_seperation_cm</span><span class="p">)]</span>
|
||||
</span><span id="electric_field_strength_two_points-81"><a href="#electric_field_strength_two_points-81"><span class="linenos"> 81</span></a> <span class="n">total_efs</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
</span><span id="electric_field_strength_two_points-82"><a href="#electric_field_strength_two_points-82"><span class="linenos"> 82</span></a> <span class="c1"># Work out how far A and B are from P (vector)</span>
|
||||
</span><span id="electric_field_strength_two_points-83"><a href="#electric_field_strength_two_points-83"><span class="linenos"> 83</span></a> <span class="k">if</span> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-84"><a href="#electric_field_strength_two_points-84"><span class="linenos"> 84</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
</span><span id="electric_field_strength_two_points-85"><a href="#electric_field_strength_two_points-85"><span class="linenos"> 85</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="electric_field_strength_two_points-86"><a href="#electric_field_strength_two_points-86"><span class="linenos"> 86</span></a> <span class="k">elif</span> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-87"><a href="#electric_field_strength_two_points-87"><span class="linenos"> 87</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
</span><span id="electric_field_strength_two_points-88"><a href="#electric_field_strength_two_points-88"><span class="linenos"> 88</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="electric_field_strength_two_points-89"><a href="#electric_field_strength_two_points-89"><span class="linenos"> 89</span></a> <span class="k">else</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-90"><a href="#electric_field_strength_two_points-90"><span class="linenos"> 90</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
|
||||
</span><span id="electric_field_strength_two_points-91"><a href="#electric_field_strength_two_points-91"><span class="linenos"> 91</span></a> <span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
</span><span id="electric_field_strength_two_points-92"><a href="#electric_field_strength_two_points-92"><span class="linenos"> 92</span></a>
|
||||
</span><span id="electric_field_strength_two_points-93"><a href="#electric_field_strength_two_points-93"><span class="linenos"> 93</span></a> <span class="c1"># Work out the EFS at point P caused by A and B seperatley, then sum them together in `total_efs`</span>
|
||||
</span><span id="electric_field_strength_two_points-94"><a href="#electric_field_strength_two_points-94"><span class="linenos"> 94</span></a> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">arrangement</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-95"><a href="#electric_field_strength_two_points-95"><span class="linenos"> 95</span></a> <span class="k">if</span> <span class="n">point</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'P'</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-96"><a href="#electric_field_strength_two_points-96"><span class="linenos"> 96</span></a> <span class="k">continue</span>
|
||||
</span><span id="electric_field_strength_two_points-97"><a href="#electric_field_strength_two_points-97"><span class="linenos"> 97</span></a> <span class="k">else</span><span class="p">:</span>
|
||||
</span><span id="electric_field_strength_two_points-98"><a href="#electric_field_strength_two_points-98"><span class="linenos"> 98</span></a> <span class="n">efs</span> <span class="o">=</span> <span class="p">((</span><span class="mf">8.99</span><span class="o">*</span><span class="mi">10</span><span class="o">**</span><span class="mi">9</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">point</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="o">**-</span><span class="mi">6</span><span class="p">))</span><span class="o">/</span><span class="p">((</span><span class="n">point</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># efs = kQ/r²</span>
|
||||
</span><span id="electric_field_strength_two_points-99"><a href="#electric_field_strength_two_points-99"><span class="linenos"> 99</span></a> <span class="k">if</span> <span class="n">point</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span> <span class="n">efs</span> <span class="o">=</span> <span class="o">-</span><span class="n">efs</span>
|
||||
</span><span id="electric_field_strength_two_points-100"><a href="#electric_field_strength_two_points-100"><span class="linenos">100</span></a> <span class="n">point</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">efs</span><span class="p">)</span>
|
||||
</span><span id="electric_field_strength_two_points-101"><a href="#electric_field_strength_two_points-101"><span class="linenos">101</span></a> <span class="n">total_efs</span> <span class="o">+=</span> <span class="n">efs</span>
|
||||
</span><span id="electric_field_strength_two_points-102"><a href="#electric_field_strength_two_points-102"><span class="linenos">102</span></a>
|
||||
</span><span id="electric_field_strength_two_points-103"><a href="#electric_field_strength_two_points-103"><span class="linenos">103</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Charges A and B and point P are arranged like this:</span><span class="se">\n</span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> <-- $</span><span class="si">{</span><span class="n">seperations</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">$ cm --> </span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2"> <-- $</span><span class="si">{</span><span class="n">seperations</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">}</span><span class="s2">$ cm --> </span><span class="si">{</span><span class="n">arrangement</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="se">\n</span><span class="s2">Where A and B have charges of $</span><span class="si">{</span><span class="n">a_charge</span><span class="si">}</span><span class="s2">$ µC and $</span><span class="si">{</span><span class="n">b_charge</span><span class="si">}</span><span class="s2">$ µC</span><span class="se">\n</span><span class="s2">What is the electric field strength at point P?"</span>
|
||||
</span><span id="electric_field_strength_two_points-104"><a href="#electric_field_strength_two_points-104"><span class="linenos">104</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">total_efs</span><span class="p">)</span><span class="si">}</span><span class="s2"> NC^</span><span class="si">{</span><span class="o">-</span><span class="mi">1</span><span class="si">}</span><span class="s2">$ (to the right)"</span>
|
||||
</span><span id="electric_field_strength_two_points-105"><a href="#electric_field_strength_two_points-105"><span class="linenos">105</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
<div class="docstring"><p>Calculate the total electric field strength at point P with given points A and B, using the equation kQ/r²</p>
|
||||
|
||||
<table>
|
||||
<thead>
|
||||
<tr>
|
||||
<th>Ex. Problem</th>
|
||||
<th>Ex. Solution</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>Charges A and B and point P are arranged like this: B <-- 7 cm --> P <-- 79 cm --> A, Where A and B have charges of -56 µC and -410 µC, What is the electric field strength at point P?</td>
|
||||
<td>$-751417824 NC^{-1}$ (to the right)</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
|
||||
|
||||
</section>
|
||||
<section id="fringe_spacing">
|
||||
<input id="fringe_spacing-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
|
||||
@@ -341,21 +478,21 @@
|
||||
|
||||
</div>
|
||||
<a class="headerlink" href="#fringe_spacing"></a>
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="fringe_spacing-69"><a href="#fringe_spacing-69"><span class="linenos">69</span></a><span class="k">def</span><span class="w"> </span><span class="nf">fringe_spacing</span><span class="p">(</span><span class="n">max_screen_distance</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
|
||||
</span><span id="fringe_spacing-70"><a href="#fringe_spacing-70"><span class="linenos">70</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the fringe spacing in a double slit experiment with w=(λD)/s</span>
|
||||
</span><span id="fringe_spacing-71"><a href="#fringe_spacing-71"><span class="linenos">71</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="fringe_spacing-72"><a href="#fringe_spacing-72"><span class="linenos">72</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="fringe_spacing-73"><a href="#fringe_spacing-73"><span class="linenos">73</span></a><span class="sd"> | A laser with a wavelength of $450nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $12m$ from the slits and the slits are $0.30mm$ apart. Calculate the spacing between the bright fringes. | Using the equation $\\frac{{\\lambda D}}{{s}}$, we get a fringe spacing of $0.018m$ |</span>
|
||||
</span><span id="fringe_spacing-74"><a href="#fringe_spacing-74"><span class="linenos">74</span></a><span class="sd"> """</span>
|
||||
</span><span id="fringe_spacing-75"><a href="#fringe_spacing-75"><span class="linenos">75</span></a> <span class="n">wavelength_nm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">380</span><span class="p">,</span><span class="mi">750</span><span class="p">)</span> <span class="c1"># Random wavelength between violet and red (nm)</span>
|
||||
</span><span id="fringe_spacing-76"><a href="#fringe_spacing-76"><span class="linenos">76</span></a> <span class="n">screen_distance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_screen_distance</span><span class="p">)</span> <span class="c1"># Random distance between screen and slits (m)</span>
|
||||
</span><span id="fringe_spacing-77"><a href="#fringe_spacing-77"><span class="linenos">77</span></a> <span class="n">slit_spacing_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="p">)</span> <span class="c1"># Random slit spacing (mm)</span>
|
||||
</span><span id="fringe_spacing-78"><a href="#fringe_spacing-78"><span class="linenos">78</span></a>
|
||||
</span><span id="fringe_spacing-79"><a href="#fringe_spacing-79"><span class="linenos">79</span></a> <span class="n">fringe_spacing</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((((</span><span class="n">wavelength_nm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span> <span class="o">*</span> <span class="n">screen_distance</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">slit_spacing_mm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">3</span><span class="p">)),</span><span class="mi">5</span><span class="p">)</span>
|
||||
</span><span id="fringe_spacing-80"><a href="#fringe_spacing-80"><span class="linenos">80</span></a>
|
||||
</span><span id="fringe_spacing-81"><a href="#fringe_spacing-81"><span class="linenos">81</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser with a wavelength of $</span><span class="si">{</span><span class="n">wavelength_nm</span><span class="si">}</span><span class="s2">nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $</span><span class="si">{</span><span class="n">screen_distance</span><span class="si">}</span><span class="s2">m$ from the slits and the slits are $</span><span class="si">{</span><span class="n">slit_spacing_mm</span><span class="si">}</span><span class="s2">mm$ apart. Calculate the spacing between the bright fringes."</span>
|
||||
</span><span id="fringe_spacing-82"><a href="#fringe_spacing-82"><span class="linenos">82</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Using the equation $</span><span class="se">\\</span><span class="s2">frac</span><span class="se">{{\\</span><span class="s2">lambda D</span><span class="se">}}{{</span><span class="s2">s</span><span class="se">}}</span><span class="s2">$, we get a fringe spacing of $</span><span class="si">{</span><span class="n">fringe_spacing</span><span class="si">}</span><span class="s2">m$"</span>
|
||||
</span><span id="fringe_spacing-83"><a href="#fringe_spacing-83"><span class="linenos">83</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="fringe_spacing-109"><a href="#fringe_spacing-109"><span class="linenos">109</span></a><span class="k">def</span><span class="w"> </span><span class="nf">fringe_spacing</span><span class="p">(</span><span class="n">max_screen_distance</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
|
||||
</span><span id="fringe_spacing-110"><a href="#fringe_spacing-110"><span class="linenos">110</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the fringe spacing in a double slit experiment with w=(λD)/s</span>
|
||||
</span><span id="fringe_spacing-111"><a href="#fringe_spacing-111"><span class="linenos">111</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="fringe_spacing-112"><a href="#fringe_spacing-112"><span class="linenos">112</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="fringe_spacing-113"><a href="#fringe_spacing-113"><span class="linenos">113</span></a><span class="sd"> | A laser with a wavelength of $450nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $12m$ from the slits and the slits are $0.30mm$ apart. Calculate the spacing between the bright fringes. | Using the equation $\\frac{{\\lambda D}}{{s}}$, we get a fringe spacing of $0.018m$ |</span>
|
||||
</span><span id="fringe_spacing-114"><a href="#fringe_spacing-114"><span class="linenos">114</span></a><span class="sd"> """</span>
|
||||
</span><span id="fringe_spacing-115"><a href="#fringe_spacing-115"><span class="linenos">115</span></a> <span class="n">wavelength_nm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">380</span><span class="p">,</span><span class="mi">750</span><span class="p">)</span> <span class="c1"># Random wavelength between violet and red (nm)</span>
|
||||
</span><span id="fringe_spacing-116"><a href="#fringe_spacing-116"><span class="linenos">116</span></a> <span class="n">screen_distance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_screen_distance</span><span class="p">)</span> <span class="c1"># Random distance between screen and slits (m)</span>
|
||||
</span><span id="fringe_spacing-117"><a href="#fringe_spacing-117"><span class="linenos">117</span></a> <span class="n">slit_spacing_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_slit_spacing_mm</span><span class="p">)</span> <span class="c1"># Random slit spacing (mm)</span>
|
||||
</span><span id="fringe_spacing-118"><a href="#fringe_spacing-118"><span class="linenos">118</span></a>
|
||||
</span><span id="fringe_spacing-119"><a href="#fringe_spacing-119"><span class="linenos">119</span></a> <span class="n">fringe_spacing</span> <span class="o">=</span> <span class="nb">round</span><span class="p">((((</span><span class="n">wavelength_nm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span> <span class="o">*</span> <span class="n">screen_distance</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">slit_spacing_mm</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**-</span><span class="mi">3</span><span class="p">)),</span><span class="mi">5</span><span class="p">)</span>
|
||||
</span><span id="fringe_spacing-120"><a href="#fringe_spacing-120"><span class="linenos">120</span></a>
|
||||
</span><span id="fringe_spacing-121"><a href="#fringe_spacing-121"><span class="linenos">121</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser with a wavelength of $</span><span class="si">{</span><span class="n">wavelength_nm</span><span class="si">}</span><span class="s2">nm$ is shone through a double slit system to produce an interference pattern on a screen. The screen is $</span><span class="si">{</span><span class="n">screen_distance</span><span class="si">}</span><span class="s2">m$ from the slits and the slits are $</span><span class="si">{</span><span class="n">slit_spacing_mm</span><span class="si">}</span><span class="s2">mm$ apart. Calculate the spacing between the bright fringes."</span>
|
||||
</span><span id="fringe_spacing-122"><a href="#fringe_spacing-122"><span class="linenos">122</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Using the equation $</span><span class="se">\\</span><span class="s2">frac</span><span class="se">{{\\</span><span class="s2">lambda D</span><span class="se">}}{{</span><span class="s2">s</span><span class="se">}}</span><span class="s2">$, we get a fringe spacing of $</span><span class="si">{</span><span class="n">fringe_spacing</span><span class="si">}</span><span class="s2">m$"</span>
|
||||
</span><span id="fringe_spacing-123"><a href="#fringe_spacing-123"><span class="linenos">123</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
@@ -378,6 +515,58 @@
|
||||
</div>
|
||||
|
||||
|
||||
</section>
|
||||
<section id="diffraction_grating_wavelength">
|
||||
<input id="diffraction_grating_wavelength-view-source" class="view-source-toggle-state" type="checkbox" aria-hidden="true" tabindex="-1">
|
||||
<div class="attr function">
|
||||
|
||||
<span class="def">def</span>
|
||||
<span class="name">diffraction_grating_wavelength</span><span class="signature pdoc-code condensed">(<span class="param"><span class="n">min_slits_per_mm</span><span class="o">=</span><span class="mi">100</span>, </span><span class="param"><span class="n">max_slits_per_mm</span><span class="o">=</span><span class="mi">500</span>, </span><span class="param"><span class="n">max_order_number</span><span class="o">=</span><span class="mi">5</span></span><span class="return-annotation">):</span></span>
|
||||
|
||||
<label class="view-source-button" for="diffraction_grating_wavelength-view-source"><span>View Source</span></label>
|
||||
|
||||
</div>
|
||||
<a class="headerlink" href="#diffraction_grating_wavelength"></a>
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="diffraction_grating_wavelength-125"><a href="#diffraction_grating_wavelength-125"><span class="linenos">125</span></a><span class="k">def</span><span class="w"> </span><span class="nf">diffraction_grating_wavelength</span><span class="p">(</span><span class="n">min_slits_per_mm</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_slits_per_mm</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_order_number</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
|
||||
</span><span id="diffraction_grating_wavelength-126"><a href="#diffraction_grating_wavelength-126"><span class="linenos">126</span></a><span class="w"> </span><span class="sa">r</span><span class="sd">"""Calculate the wavelength when given the number of slits per mm, order number and angle of order using the equation nλ = dsinθ</span>
|
||||
</span><span id="diffraction_grating_wavelength-127"><a href="#diffraction_grating_wavelength-127"><span class="linenos">127</span></a>
|
||||
</span><span id="diffraction_grating_wavelength-128"><a href="#diffraction_grating_wavelength-128"><span class="linenos">128</span></a><span class="sd"> | Ex. Problem | Ex. Solution |</span>
|
||||
</span><span id="diffraction_grating_wavelength-129"><a href="#diffraction_grating_wavelength-129"><span class="linenos">129</span></a><span class="sd"> | --- | --- |</span>
|
||||
</span><span id="diffraction_grating_wavelength-130"><a href="#diffraction_grating_wavelength-130"><span class="linenos">130</span></a><span class="sd"> | A laser is shone through a diffraction grating which has $293$ lines per mm, the fringe of order number $2$ is at an angle of $0.39$ rad. Calculate the wavelength of the light | $\lambda = 6.487856913364529e-07m = 649nm |</span>
|
||||
</span><span id="diffraction_grating_wavelength-131"><a href="#diffraction_grating_wavelength-131"><span class="linenos">131</span></a>
|
||||
</span><span id="diffraction_grating_wavelength-132"><a href="#diffraction_grating_wavelength-132"><span class="linenos">132</span></a><span class="sd"> """</span>
|
||||
</span><span id="diffraction_grating_wavelength-133"><a href="#diffraction_grating_wavelength-133"><span class="linenos">133</span></a> <span class="n">slits_per_mm</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">min_slits_per_mm</span><span class="p">,</span> <span class="n">max_slits_per_mm</span><span class="p">)</span>
|
||||
</span><span id="diffraction_grating_wavelength-134"><a href="#diffraction_grating_wavelength-134"><span class="linenos">134</span></a> <span class="n">slit_spacing</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">slits_per_mm</span> <span class="o">*</span> <span class="mi">1000</span><span class="p">)</span>
|
||||
</span><span id="diffraction_grating_wavelength-135"><a href="#diffraction_grating_wavelength-135"><span class="linenos">135</span></a> <span class="n">order_number</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_order_number</span><span class="p">)</span>
|
||||
</span><span id="diffraction_grating_wavelength-136"><a href="#diffraction_grating_wavelength-136"><span class="linenos">136</span></a> <span class="n">angle_of_order</span> <span class="o">=</span> <span class="nb">round</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="mf">0.2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
|
||||
</span><span id="diffraction_grating_wavelength-137"><a href="#diffraction_grating_wavelength-137"><span class="linenos">137</span></a> <span class="n">wavelength</span> <span class="o">=</span> <span class="p">((</span><span class="n">slit_spacing</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle_of_order</span><span class="p">))</span> <span class="o">/</span> <span class="n">order_number</span><span class="p">)</span>
|
||||
</span><span id="diffraction_grating_wavelength-138"><a href="#diffraction_grating_wavelength-138"><span class="linenos">138</span></a>
|
||||
</span><span id="diffraction_grating_wavelength-139"><a href="#diffraction_grating_wavelength-139"><span class="linenos">139</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"A laser is shone through a diffraction grating which has $</span><span class="si">{</span><span class="n">slits_per_mm</span><span class="si">}</span><span class="s2">$ lines per mm, the fringe of order number $</span><span class="si">{</span><span class="n">order_number</span><span class="si">}</span><span class="s2">$ is at an angle of $</span><span class="si">{</span><span class="n">angle_of_order</span><span class="si">}</span><span class="s2">$ rad. Calculate the wavelength of the light"</span>
|
||||
</span><span id="diffraction_grating_wavelength-140"><a href="#diffraction_grating_wavelength-140"><span class="linenos">140</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="se">\\</span><span class="s2">lambda = </span><span class="si">{</span><span class="n">wavelength</span><span class="si">}</span><span class="s2">m = </span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">wavelength</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">10</span><span class="o">**-</span><span class="mi">9</span><span class="p">)</span><span class="si">}</span><span class="s2">nm$"</span>
|
||||
</span><span id="diffraction_grating_wavelength-141"><a href="#diffraction_grating_wavelength-141"><span class="linenos">141</span></a>
|
||||
</span><span id="diffraction_grating_wavelength-142"><a href="#diffraction_grating_wavelength-142"><span class="linenos">142</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
<div class="docstring"><p>Calculate the wavelength when given the number of slits per mm, order number and angle of order using the equation nλ = dsinθ</p>
|
||||
|
||||
<table>
|
||||
<thead>
|
||||
<tr>
|
||||
<th>Ex. Problem</th>
|
||||
<th>Ex. Solution</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>A laser is shone through a diffraction grating which has $293$ lines per mm, the fringe of order number $2$ is at an angle of $0.39$ rad. Calculate the wavelength of the light</td>
|
||||
<td>$\lambda = 6.487856913364529e-07m = 649nm</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
|
||||
|
||||
</section>
|
||||
</main>
|
||||
<script>
|
||||
|
||||
@@ -96,7 +96,7 @@
|
||||
|
||||
<div class="pdoc-code codehilite"><pre><span></span><span id="L-1"><a href="#L-1"><span class="linenos"> 1</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">random</span>
|
||||
</span><span id="L-2"><a href="#L-2"><span class="linenos"> 2</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">math</span>
|
||||
</span><span id="L-3"><a href="#L-3"><span class="linenos"> 3</span></a>
|
||||
</span><span id="L-3"><a href="#L-3"><span class="linenos"> 3</span></a><span class="kn">import</span><span class="w"> </span><span class="nn">scipy.stats</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">stats</span>
|
||||
</span><span id="L-4"><a href="#L-4"><span class="linenos"> 4</span></a>
|
||||
</span><span id="L-5"><a href="#L-5"><span class="linenos"> 5</span></a><span class="k">def</span><span class="w"> </span><span class="nf">combinations</span><span class="p">(</span><span class="n">max_lengthgth</span><span class="o">=</span><span class="mi">20</span><span class="p">):</span>
|
||||
</span><span id="L-6"><a href="#L-6"><span class="linenos"> 6</span></a><span class="w"> </span><span class="sd">"""Combinations of Objects</span>
|
||||
@@ -274,6 +274,25 @@
|
||||
</span><span id="L-178"><a href="#L-178"><span class="linenos">178</span></a>
|
||||
</span><span id="L-179"><a href="#L-179"><span class="linenos">179</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Number of Permutations from $</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s2">$ objects picked $</span><span class="si">{</span><span class="n">b</span><span class="si">}</span><span class="s2">$ at a time is: "</span>
|
||||
</span><span id="L-180"><a href="#L-180"><span class="linenos">180</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">solution</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="L-181"><a href="#L-181"><span class="linenos">181</span></a>
|
||||
</span><span id="L-182"><a href="#L-182"><span class="linenos">182</span></a><span class="c1"># TODO</span>
|
||||
</span><span id="L-183"><a href="#L-183"><span class="linenos">183</span></a><span class="c1">#def normal_distribution_bounds(max_mean=100, max_variance=10): # max value for mean is absolute</span>
|
||||
</span><span id="L-184"><a href="#L-184"><span class="linenos">184</span></a><span class="w"> </span><span class="sd">"""</span>
|
||||
</span><span id="L-185"><a href="#L-185"><span class="linenos">185</span></a>
|
||||
</span><span id="L-186"><a href="#L-186"><span class="linenos">186</span></a>
|
||||
</span><span id="L-187"><a href="#L-187"><span class="linenos">187</span></a><span class="sd"> """</span>
|
||||
</span><span id="L-188"><a href="#L-188"><span class="linenos">188</span></a> <span class="c1"># P(x1 <= X <= x2) = CDF(x2) − CDF(x1)</span>
|
||||
</span><span id="L-189"><a href="#L-189"><span class="linenos">189</span></a>
|
||||
</span><span id="L-190"><a href="#L-190"><span class="linenos">190</span></a> <span class="c1"># X(x) = 1/2[1+erf(x−μ√2σ)]</span>
|
||||
</span><span id="L-191"><a href="#L-191"><span class="linenos">191</span></a> <span class="n">mean</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_mean</span><span class="p">,</span> <span class="n">max_mean</span><span class="p">)</span>
|
||||
</span><span id="L-192"><a href="#L-192"><span class="linenos">192</span></a> <span class="n">variance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_variance</span><span class="p">)</span>
|
||||
</span><span id="L-193"><a href="#L-193"><span class="linenos">193</span></a> <span class="n">bound_1</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">-</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))),</span> <span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))))</span>
|
||||
</span><span id="L-194"><a href="#L-194"><span class="linenos">194</span></a> <span class="n">bound_2</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">bound_1</span><span class="p">,</span> <span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))))</span>
|
||||
</span><span id="L-195"><a href="#L-195"><span class="linenos">195</span></a> <span class="n">answer</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">bound_2</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">variance</span><span class="p">)</span> <span class="o">-</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">bound_1</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">variance</span><span class="p">)</span>
|
||||
</span><span id="L-196"><a href="#L-196"><span class="linenos">196</span></a>
|
||||
</span><span id="L-197"><a href="#L-197"><span class="linenos">197</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"What is the area under the Normal Distribution $X~N(</span><span class="si">{</span><span class="n">mean</span><span class="si">}</span><span class="s2">,</span><span class="si">{</span><span class="n">variance</span><span class="si">}</span><span class="s2">)$ between $</span><span class="si">{</span><span class="n">bound_1</span><span class="si">}</span><span class="s2">$ and $</span><span class="si">{</span><span class="n">bound_2</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="L-198"><a href="#L-198"><span class="linenos">198</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">answer</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="L-199"><a href="#L-199"><span class="linenos">199</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
@@ -657,6 +676,25 @@
|
||||
</span><span id="permutation-179"><a href="#permutation-179"><span class="linenos">179</span></a>
|
||||
</span><span id="permutation-180"><a href="#permutation-180"><span class="linenos">180</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Number of Permutations from $</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s2">$ objects picked $</span><span class="si">{</span><span class="n">b</span><span class="si">}</span><span class="s2">$ at a time is: "</span>
|
||||
</span><span id="permutation-181"><a href="#permutation-181"><span class="linenos">181</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">solution</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="permutation-182"><a href="#permutation-182"><span class="linenos">182</span></a>
|
||||
</span><span id="permutation-183"><a href="#permutation-183"><span class="linenos">183</span></a><span class="c1"># TODO</span>
|
||||
</span><span id="permutation-184"><a href="#permutation-184"><span class="linenos">184</span></a><span class="c1">#def normal_distribution_bounds(max_mean=100, max_variance=10): # max value for mean is absolute</span>
|
||||
</span><span id="permutation-185"><a href="#permutation-185"><span class="linenos">185</span></a><span class="w"> </span><span class="sd">"""</span>
|
||||
</span><span id="permutation-186"><a href="#permutation-186"><span class="linenos">186</span></a>
|
||||
</span><span id="permutation-187"><a href="#permutation-187"><span class="linenos">187</span></a>
|
||||
</span><span id="permutation-188"><a href="#permutation-188"><span class="linenos">188</span></a><span class="sd"> """</span>
|
||||
</span><span id="permutation-189"><a href="#permutation-189"><span class="linenos">189</span></a> <span class="c1"># P(x1 <= X <= x2) = CDF(x2) − CDF(x1)</span>
|
||||
</span><span id="permutation-190"><a href="#permutation-190"><span class="linenos">190</span></a>
|
||||
</span><span id="permutation-191"><a href="#permutation-191"><span class="linenos">191</span></a> <span class="c1"># X(x) = 1/2[1+erf(x−μ√2σ)]</span>
|
||||
</span><span id="permutation-192"><a href="#permutation-192"><span class="linenos">192</span></a> <span class="n">mean</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">max_mean</span><span class="p">,</span> <span class="n">max_mean</span><span class="p">)</span>
|
||||
</span><span id="permutation-193"><a href="#permutation-193"><span class="linenos">193</span></a> <span class="n">variance</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_variance</span><span class="p">)</span>
|
||||
</span><span id="permutation-194"><a href="#permutation-194"><span class="linenos">194</span></a> <span class="n">bound_1</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">-</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))),</span> <span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))))</span>
|
||||
</span><span id="permutation-195"><a href="#permutation-195"><span class="linenos">195</span></a> <span class="n">bound_2</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">bound_1</span><span class="p">,</span> <span class="nb">round</span><span class="p">(</span><span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance</span><span class="p">))))</span>
|
||||
</span><span id="permutation-196"><a href="#permutation-196"><span class="linenos">196</span></a> <span class="n">answer</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">bound_2</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">variance</span><span class="p">)</span> <span class="o">-</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">bound_1</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">variance</span><span class="p">)</span>
|
||||
</span><span id="permutation-197"><a href="#permutation-197"><span class="linenos">197</span></a>
|
||||
</span><span id="permutation-198"><a href="#permutation-198"><span class="linenos">198</span></a> <span class="n">problem</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"What is the area under the Normal Distribution $X~N(</span><span class="si">{</span><span class="n">mean</span><span class="si">}</span><span class="s2">,</span><span class="si">{</span><span class="n">variance</span><span class="si">}</span><span class="s2">)$ between $</span><span class="si">{</span><span class="n">bound_1</span><span class="si">}</span><span class="s2">$ and $</span><span class="si">{</span><span class="n">bound_2</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="permutation-199"><a href="#permutation-199"><span class="linenos">199</span></a> <span class="n">solution</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"$</span><span class="si">{</span><span class="n">answer</span><span class="si">}</span><span class="s2">$"</span>
|
||||
</span><span id="permutation-200"><a href="#permutation-200"><span class="linenos">200</span></a> <span class="k">return</span> <span class="n">problem</span><span class="p">,</span> <span class="n">solution</span>
|
||||
</span></pre></div>
|
||||
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user