Question
A uniform but time-varying magnetic field $$B\left( t \right)$$ exists in a circular region of radius $$a$$ and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point $$P$$ at a distance $$r$$ from the centre of the circular region
A uniform but time-varying magnetic field $$B\left( t \right)$$ exists in a circular region of radius $$a$$ and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point $$P$$ at a distance $$r$$ from the centre of the circular region
A.
is zero
B.
decreases as $$\frac{1}{r}$$
C.
increases as $$r$$
D.
decreases as $$\frac{1}{{{r^2}}}$$
Answer :
decreases as $$\frac{1}{r}$$
Solution :
$$\eqalign{ & \oint {\overrightarrow E } .\overrightarrow {d\ell } = \frac{{d\phi }}{{dt}} = \frac{d}{{dt}}\left( {\overrightarrow B .\overrightarrow A } \right) = \frac{d}{{dt}}\left( {BA\cos {0^ \circ }} \right) = A\frac{{dB}}{{dt}} \cr & \Rightarrow E\left( {2\pi r} \right) = \pi {a^2}\frac{{dB}}{{dt}}{\text{ for }}r \geqslant a \cr & \Rightarrow E = \frac{{{a^2}}}{{2r}}\frac{{dB}}{{dt}} \Rightarrow E \propto \frac{1}{r} \cr} $$
$$\eqalign{ & \oint {\overrightarrow E } .\overrightarrow {d\ell } = \frac{{d\phi }}{{dt}} = \frac{d}{{dt}}\left( {\overrightarrow B .\overrightarrow A } \right) = \frac{d}{{dt}}\left( {BA\cos {0^ \circ }} \right) = A\frac{{dB}}{{dt}} \cr & \Rightarrow E\left( {2\pi r} \right) = \pi {a^2}\frac{{dB}}{{dt}}{\text{ for }}r \geqslant a \cr & \Rightarrow E = \frac{{{a^2}}}{{2r}}\frac{{dB}}{{dt}} \Rightarrow E \propto \frac{1}{r} \cr} $$
