First slide

Induced Electric Field and Magnetic field

Question

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

Moderate

A

Is zero
B

Decreases with increase in r
C

Increases with increase in r
D

Decreases with decrease in r

Solution

Construct a concentric circle of radius r. The induced electric field (E) at any point on the circle is equal to that at P. For this circle
$∮ \vec{E} \cdot d \vec{l} = |\frac{d ϕ}{d t}| = A |\frac{d B}{d t}|$
$or E \times (2 π r) = π a^{2} \cdot |\frac{d B}{d t}|$
$\Rightarrow E = \frac{a^{2}}{2 r} |\frac{d B}{d t}| \Rightarrow E \propto \frac{1}{r}$

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