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

• A. is Zero
• B. decreases as 1/r
• C. increases ac r
• D. decreases as $1/r^2$

Answer 1

Answer: (B)

decreases as 1/r

Answer 2

$\int E \cdot dI = \bigg| \frac{d\phi}{dt} \bigg| = S \bigg|\frac{dB}{dt} \bigg|$ or $\, \, \, \, \, \, \, E(2 \pi r) = \pi a^2 \bigg|\frac{dB}{dt} \bigg|$ $For \, r \ge a,$ $\therefore \, \, \, \, \, \, \, \, \, \, \, \, E = \frac{a^2}{2r} \bigg|\frac{dB}{dt} \bigg|$ $\therefore$ Induced electric field $\propto 1 /r$ $For \, r \le a,$ $\, \, \, \, \, \, \, \, \, \, E(2\pi r) = \pi r^2 \bigg| \frac{dB}{dt} \bigg|$ or $\, \, \, \, \, \, \, \, \, \, \, \, \, \, E = \frac{r}{2} \bigg| \frac{dB}{dt} \bigg|$ ot $\, \, \, \, \, \, \, \, \, \, \, \, \, \, E \propto r$ $At \, \, \, \, \, \, \, \, \, \, r = a , E = \frac{a}{2} \bigg| \frac{dB}{dt} \bigg|$ Therefore, variation of E with r (distance from centre) will be as follows: