Question
A uniform magnetic field is restricted within a region of radius $$r.$$ The magnetic field changes with time at a rate $$\frac{{dB}}{{dt}}.$$ 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 emf generated is
A uniform magnetic field is restricted within a region of radius $$r.$$ The magnetic field changes with time at a rate $$\frac{{dB}}{{dt}}.$$ 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 emf generated is
A.
zero in loop 1 and zero in loop 2
B.
$$ - \frac{{dB}}{{dt}}\pi {r^2}$$ in loop 1 and $$ - \frac{{dB}}{{dt}}\pi {r^2}$$ in loop 2
C.
$$ - \frac{{dB}}{{dt}}\pi {R^2}$$ in loop 1 and zero in loop 2
D.
$$ - \frac{{dB}}{{dt}}\pi {r^2}$$ in loop 1 and zero in loop 2
Answer :
$$ - \frac{{dB}}{{dt}}\pi {R^2}$$ in loop 1 and zero in loop 2
Solution :
Induced emf in the region is given by
$$\left| e \right| = \frac{{d\phi }}{{dt}}$$
where, $$\phi = BA = \pi {r^2}B$$
$$ \Rightarrow \frac{{d\phi }}{{dt}} = - \pi {r^2}\frac{{dB}}{{dt}}$$
Rate of change of magnetic flux associated with loop 1
$${e_1} = - \frac{{d{\phi _1}}}{{dt}} = - \pi {r^2}\frac{{dB}}{{dt}}$$
Similarly $${e_2} = $$ emf associated with loop 2
$$ = - \frac{{d{\phi _2}}}{{dt}} = 0\,\,\left[ {\because {\phi _2} = 0} \right]$$
Induced emf in the region is given by
$$\left| e \right| = \frac{{d\phi }}{{dt}}$$
where, $$\phi = BA = \pi {r^2}B$$
$$ \Rightarrow \frac{{d\phi }}{{dt}} = - \pi {r^2}\frac{{dB}}{{dt}}$$
Rate of change of magnetic flux associated with loop 1
$${e_1} = - \frac{{d{\phi _1}}}{{dt}} = - \pi {r^2}\frac{{dB}}{{dt}}$$
Similarly $${e_2} = $$ emf associated with loop 2
$$ = - \frac{{d{\phi _2}}}{{dt}} = 0\,\,\left[ {\because {\phi _2} = 0} \right]$$
