7677
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<p style="text-align: justify;">An electric kettle is a quite simple household appliance which is used everyday to convert electrical energy to heat energy. The diagram shows a circuit which provides a simple model for what happens in the electric kettle. </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: center;"><mdo:image alt="" src="circuitt.bmp" style="width: 358px; height: 206px; float: left;"></mdo:image> <mdo:image alt="Kettle" src="http://www.sciencemuseum.org.uk/images/object_images/535x535/10325939.jpg" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default; width: 250px; height: 180px;"></mdo:image></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: left;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">The kettle diagram: V is the voltage of the power supply (a power supply in UK has 240 volts (V)), R is the resistance of a heating element measured in ohms ($\Omega$), I is a current flowing around the circuit in
amps (A).</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">The experiment was carried out to investigate the relationship between the resistance of heating element and the temperature which settle down after a while. Results are shown in the table.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<table border="1" cellpadding="10" cellspacing="0" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; font-size: 0.95em;" summary="resistance-temperature data">
<tbody align="center" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;"><strong style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">Resistance R/$\Omega$</strong></td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;"><strong style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">Temperature T/°C</strong></td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">65</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">97</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">100</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">70</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">150</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">53</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">200</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">45</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">250</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;">40</td>
</tr>
</tbody>
</table>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
</div>
<ol style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 40px; padding-bottom: 1ex; padding-left: 2em;">
<li style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Why does the temperature settle down after a while?</li>
<li style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Plot the graph of data for the temperature against the resistance.</li>
<li style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Could you recongise the function? Using points (100, 70) and (250, 40) find the equation of this function. (Hint: the function is of the form T = T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> + C/R where T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> and C are constants.)</li>
<li style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Find the temperature of the room where the experiment was conducted.</li>
<li style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Does this equation is suitable for all values of resistance?</li>
</ol>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">In practise we know that for an electric kettle to boil water takes 3-4 minutes. Could you</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">investigate how the time needed to boil water is related with the amount of water which are you heating?</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;"> </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Extension: the power of the kettle is given by the formula where R is the resistance of the heating element, r is the internal resistance and V is the voltage of power supply. $$P = \frac{RV^2}{(r+R)^2}$$
Suppose it is given that V = 240 volts, r = 5 ohms. Find R such that we get the maximum power.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Try to deduce generally the value of R for which the power is the biggest if the internal resistance is r.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;"> </p>
<hr></hr>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: left;"><em>An old electric kettle picture is taken from http://www.sciencemuseum.org.uk/images/I059/10325939.aspx</em></p>
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<mdoxml version="1.0"><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">1. The temperature in the kettle settles down because dissipated heat in the heating element is equal to heat which is lost through walls of the kettle.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: left;">2.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: center;"> <mdo:image src="temp_graph.jpg" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default; width: 600px; height: 323px;"></mdo:image></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;"> </p>
<hr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"></hr>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">3. It is a graph of a reciprocal function. The function is of the form T = T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> + C/R. We use two given points (100, 70), (250, 40) to get two equations:</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">70 = T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> + C/100 (1)</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">40 = T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> + C/250
(2)</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Subtracting (2) from (1) we have 70 - 40 = C/100 - C/250 which gives C = 5000 and using (1) we have that T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub>= 70-50 = 20. Hence we have the function T = 20 + 5000/R.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">4. To find the temperature of the room we need to model the system such that a heating element is not being heated. This means we let R be sufficiently large because to make R really small is not realistic.
Thus, from the previous equation the temperature of the room is T<sub style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">0</sub> = 20.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">5. We notice that as the resistance is small the temperature is really high. Above a water boiling temperature, this means that there would be vapor. So, this equation is not suitable for small values of
resistance. We also should take into consideration the resistance of wires.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">6. It should be related linearly by the definition of specific heat capacity. </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Extension. </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Method 1. Try to make a table noting that the voltage is constant and it only scales the power.</p>
<table style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; font-size: 0.95em; width: 500px;border-spacing:1px;" border="1">
<tbody style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">R/$\Omega$</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">$R/(r+R)^2$$\times \Omega$</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">1</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0278</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">2</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0408</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">3</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0469</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">4</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0494</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">5</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0500</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">6</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0496</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">7</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0486</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">8</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0473</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">9</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0459</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">10</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.0444</td>
</tr>
</tbody>
</table>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">We see that a maximum is reached is near 5 $\Omega$. Lets make another table to explore values which are close to 5 $\Omega$.</p>
<table style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; font-size: 0.95em; width: 500px;border-spacing:1px;" border="1">
<tbody style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;padding:1px;">R/$\Omega$</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px;padding:1px;">$R/(r+R)^2$$\times \Omega$</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">4.8</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.049979</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">4.9</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.049995</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">5.0</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.050000</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">5.1</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.049995</td>
</tr>
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">5.2</td>
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; text-align: center;padding:1px;">0.049981</td>
</tr>
</tbody>
</table>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Thus, we conclude that we have a maximal power when R = 5 $\Omega$. It is equal to</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">$P = 5\times 240^2 : (5+5)^2 = 2880 W$. </p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Method 2. Try to plot the function and find the maximum by drawing the line which is perpendicular to x-axis and then estimate the resistance at that point.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<hr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"></hr>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: center;"><mdo:image alt="" src="maximal_power1.jpg" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default; width: 600px; height: 323px;"></mdo:image></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"> </p>
<hr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"></hr>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Method 3. Write the function as a quadratic equation with respect to R:</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">$$R^2+ (2r - \frac{V^2}{P})R + r^2 = 0$$ The critical point will correspond to a discriminant being zero. Thus $$D = (2r - \frac{V^2}{P})^2 - 4r^2 = 0$$ We get that $$P = \frac{V^2}{4r}$$ Plug back to the
equation the expression of P to get that $(R-r)^2 = 0$. So, R = r = 5 $\Omega$.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: justify;">Generally we have that the power is maximal when the external resistance is equal to the internal resistance. R = r and $P = \frac{V^2}{4r}$.</p>
<p style="margin: 0px; padding: 1ex 0px 0px; text-align: justify;"> </p>
<p style="margin: 0px; padding: 1ex 0px 0px; text-align: justify;">Method 4. Straight differentiation: finding the maximum point ($\frac{dP}{dR}$ = $0$)</p>
<p style="margin: 0px; padding: 1ex 0px 0px; text-align: justify;">$P$ = $\frac{RV^{2}}{(r+R)^{2}}$</p>
<p style="margin: 0px; padding: 1ex 0px 0px; text-align: justify;">By using the chain rule:</p>
<p style="margin: 0px; padding: 1ex 0px 0px; text-align: justify;">$\frac{d(vu)}{dy}$ = $v\frac{du}{dy} + u\frac{dv}{dy}$</p>
<div><br></br>
Say that $P$ = $A \times B$ where $A$ = $RV^{2}$ and $B$ = $\frac{1}{(r+R)^{2}}$</div>
<div> </div>
<div>$\frac{dP}{dR}$ = $A \times \frac{dB}{dy} + B \times \frac{dA}{dy}$ = $0$</div>
<div> </div>
<div> = $RV^{2} \times \frac{-2}{(r+R)^3} + \frac{1}{(r+R)^{2}} \times V^{2}$</div>
<div> </div>
<div> = $\frac{-2RV^{2}}{(r+R)^3} + \frac{V^{2}}{(r+R)^{2}}$</div>
<div> </div>
<div> = $-2RV^{2} + V^{2}(r+R)$</div>
<div> </div>
<div> = $V^{2}r - V^{2}R$ = $0$</div>
<div> </div>
<div>So $r$ = $R$ = $5\Omega$</div>
<div> </div>
<div>So maximum power $P$ = $\frac{RV^{2}}{(r+R)^{2}}$</div>
<div> = $\frac{V^2}{4r}$</div>
<div> </div>
<div> </div>
<div> </div>
<div> </div>
<div> </div></mdoxml>
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