Question

A wire in a shape of parabola, $y=4x^2$ is located in a uniform magnetic field $B$ which is perpendicular to the $xy$ plane. At $t=0$, an infinitely long wire starts translation from the parabola vertex with constant acceleration $a$. The induced emf $e$ in the loop, thus formed as a function of $y$ is :

• A. $e=\sqrt{2a}By$
• B. $e=\sqrt{\frac{a}{2}}By$
• C. $e=\sqrt{3a}By$
• D. $e=\sqrt{\frac{a}{3}}By$

Answer 1

The correct option is A $e=\sqrt{2a}By$


Velocity of the wire after travelling distance $y$ is,

$v^2-u^2=2as$

$\Rightarrow v^2=0^2+2ay$

$\Rightarrow v=\sqrt{2ay}$

Now, emf induced in the loop is,

$e=Bvl$

$\Rightarrow e=B\times \sqrt{2ay}\times 2x[\because l=2x]$

$\Rightarrow e=B\times \sqrt{2ay}\times 2\times \frac{\sqrt{y}}{2}[\because y=4x^2]$

$\Rightarrow e=\sqrt{2a}By$