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Question 7.53: The current in a long solenoid is increasing linearly with t...

The current in a long solenoid is increasing linearly with time, so the flux is proportional to t: = αt. Two voltmeters are connected to diametrically opposite points (A and B), together with resistors \left(R_{1} \text { and } R_{2}\right), as shown in Fig. 7.55. What is the reading on each voltmeter? Assume that these are ideal voltmeters that draw negligible current (they have huge internal resistance), and that a voltmeter registers -\int_{a}^{b} E \cdot d l between the terminals and through the meter. [Answer: V_{1}=\alpha R_{1} /\left(R_{1}+R_{2}\right) ; V_{2}=-\alpha R_{2} /\left(R_{1}+R_{2}\right) . \text { Notice that } V_{1} \neq V_{2} , even though they are connected to the same pointss !^{32}]

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ε =-\frac{d \Phi}{d t}=-\alpha . So the current in R_{1} \text { and } R_{2} \text { is } I=\frac{\alpha}{R_{1}+R_{2}} ; by Lenz’s law, it flows counterclockwise.

Now the voltage across R_{1} (which voltmeter #1 measures) is V_{1}=I R_{1}=\frac{\alpha R_{1}}{R_{1}+R_{2}} ( V_{b} is the higher potential),

and V_{2}=-I R_{2}=\frac{-\alpha R_{2}}{R_{1}+R_{2}}\left(V_{b} \text { is lower }\right) .

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