Question

As shown in the above figure, a loop surrounds three regions of the magnetic field where the magnitude of the magnetic field is decreasing at a constant rate $\alpha $ . Take the area of each region as $A$ . Now find the $\oint\overset{ \rightarrow }{E}.\overset{ \rightarrow }{d l}$ along the given loop, where $\overset{ \rightarrow }{E}$ is the induced electric field.

• A. $\alpha A\,$
• B. $-\alpha A\,$
• C. $3\,\alpha A\,$
• D. $-3\alpha \,A$

Answer 1

Answer: (B)

From faraday law of emf, a time varying magnetic field produces electric field according to the following equation:
$\oint \vec{E} \cdot \overrightarrow{d l}=-\frac{d \phi}{d t}$
also, $\phi=B . A$ is the magnetic flux assosiated substituting this in the above equation,we get:
$\Rightarrow \oint \vec{E} \cdot d \vec{l}=-\frac{d(B \cdot A)}{d t} $
$\Rightarrow \oint \vec{E} \cdot d \vec{l}=-A \frac{d B}{d t}$
Here it is given that $\frac{d B}{d t}=\alpha$ and now taking the sign of flux according to right hand curl rule we get:
$\Rightarrow \oint \vec{E} \cdot d \vec{l}=-[-(-\alpha A)-(-\alpha A)+(-\alpha A)] $
$\Rightarrow \oint \vec{E} \cdot d \vec{l}=-\alpha A$