Electromagnetic Induction and Alternating Current 5 Question 1
1. A uniform but time-varying magnetic field $B(t)$ 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
$(2000,2 M)$
(a) is zero
(b) decreases as $1 / r$
(c) increases as $r$
(d) decreases as $1 / r^{2}$
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Answer:
Correct Answer: 1. (b)
Solution:
$$ \int \mathbf{E} \cdot d \boldsymbol{l}=\frac{d \varphi}{d t}=S \frac{d B}{d t} $$
$$ \text { or } \quad E(2 \pi r)=\pi a^{2} \frac{d B}{d t} $$
For $r \geq a$,
$$ \therefore \quad E=\frac{a^{2}}{2 r} \frac{d B}{d t} $$
$\therefore$ Induced electric field $\propto 1 / r$
For $r \leq a$,
$$ \begin{array}{rlrl} & E(2 \pi r) & =\pi r^{2} \frac{d B}{d t} \\ \text { or } & E & =\frac{r}{2} \frac{d B}{d t} \\ & \text { or } & E & \propto r \\ \text { At } r=a, & E & =\frac{a}{2} \frac{d B}{d t} \end{array} $$
Therefore, variation of $E$ with $r$ (distance from centre) will be as follows
