A long solenoid of the cross-sectional radius R has a thin insulated wire ring tightly put on its winding. One half of the ring has the resistance 10 times that of the other half. The magnetic inducti

Question

A long solenoid of the cross-sectional radius R has a thin insulated wire ring tightly put on its winding. One half of the ring has the resistance 10 times that of the other half. The magnetic induction produced by the solenoid varies with time as B = b t , whose b is a constant. If the magnitude of the electric field strength in the ring is p q R b , where p and q are the smallest positive integers, then what is the value of p + q ?

TestHub · Accepted Answer

Both upper half and lower half will have same effective area of $\frac{π R^{2}}{2}$ so charge in flux will be same and induced emf will have some value. But since the resistance is different due to which current must be different but ring is as a whole is closed circuit so electric field will be generated to make the current flow in both parts to be same.

$ε + E π R - i 10 r = 0 \dots (i)$

$ε - E π R - i r = 0 \dots (i i)$

$\Rightarrow 2 E π R - 9 i r = 0$

$E = \frac{9 i r}{2 π R} \Rightarrow i = \frac{E}{11 r} = \frac{π R^{2} b}{11 r}$

$\Rightarrow E = \frac{9 r}{2 π R} \times \frac{π R^{2} b}{11 r} = \frac{9}{22} Rb$

∴ $\frac{9}{22} Rb$

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