Question

A uniform magnetic field is restricted within a region of radius $r$. The magnetic field changes with time at a rate $\frac{dB}{dt}$. The circular region containing magnetic field has a radius $r$. Loop $1$ of radius $R>>r$ encloses the region $r$ and Loop $2$ of radius $R$ is outside the region of magnetic field as shown in the figure. Then the e.m.f generated is,

• A. Zero in loop $1$ and zero in loop $2$
• B. $-\frac{\overrightarrow{dB}}{dt}\pi r^2$ in loop $1$ and $-\frac{\overrightarrow{dB}}{dt}\pi r^2$ in loop $2$
• C. $-\frac{\overrightarrow{dB}}{dt}\pi R^2$ in loop $1$, zero in loop $2$
• D. $-\frac{\overrightarrow{dB}}{dt}\pi r^2$ in loop $1$, zero in loop $2$

Answer 1

The correct option is D $-\frac{\overrightarrow{dB}}{dt}\pi r^2$ in loop $1$, zero in loop $2$

E.M.F generated in loop $1$,

$E=-\frac{d\varphi }{dt}=-\frac{d}{dt}(\overrightarrow{B}.\overrightarrow{A})=-A\frac{dB}{dt}$

The effective area, where the magnetic field exists is $A=\pi r^2$

$E_1=-\left(\pi r^2\frac{dB}{dt}\right)$

Now, as none of the magnetic field lines are linked with the loop $2$, we have,

$E_2=0$

Hence, option $\left(D\right)$ is the correct answer.