Question

A highly conducting ring of radius R is perpendicular to and concentric with the axis of a long solenoid as shown in fig. The ring has a narrow gap of width d in its circumference. The solenoid has cross sectional area A and a uniform internal field of magnitude $B_0$. Now beginning at t = 0, the solenoid current is steadily increased so that the field magnitude at any time t is given by $B\left(t\right)=B_0+\alpha twhere\alpha >0$ . Assuming that no charge can flow across the gap, the end of ring which has excess of positive charge and the magnitude of induced e.m.f. in the ring are respectively

• A. $X,A\alpha$
  
• B. $X\pi R^2\alpha$
  
• C. $Y,\pi A^2\alpha$
  
• D. $Y,\pi R^2\alpha$

Answer 1

The correct option is A $X,A\alpha$


Since the current is increasing, inward magnetic flux linked with the ring also increases (as viewed from left side). Hence induced current in the ring is anticlockwise, so end x will be positive.
Induced emf $\left|e\right|=A\frac{dB}{dt}=A\frac{d}{dt}(B_0+\alpha t)\Rightarrow \left|e\right|=A\alpha$