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An infinitely long solenoid passes through the circuit as shown. The magnetic field of the solenoid, directed into the plane of the page, is weakening which produces a constant emf of magnitude E for a closed loop outside of the solenoid. Once equilibrium is established in this circuit, the potential difference across the Switch S is $\frac{y E}{3} .$ Then find the value of y.

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$i = \frac{E}{3 R}, V_{s} = E + i R = E + \frac{E}{3} \Rightarrow V_{s} = \frac{4 E}{3}$

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An infinitely long solenoid passes through the circuit as shown. The magnetic field of the solenoid, directed into the plane of the page, is weakening which produces a constant emf of magnitude E for a closed loop outside of the solenoid. Once equilibrium is established in this circuit, the potential difference across the Switch S is yE3. Then find the value of y.