Question

A magnetic field directed along Z axis varies as $B=B_{0}x/a$, where $B_0=2tesla$ and $a=1m.$ A conducting square loop of side $\ell =\frac{1}{2}m$ is placed with its edges parallel to X and Y axes. If the loop is made to move with a constant velocity $v_0=6m/s$ directed along X axis, the induced emf (in volts) in the loop is :

Answer 1

$d\varphi =\frac{B_{0}x}{a}\times \ell dx$

$\varphi =\frac{B_0\ell }{a}\int xdx$

$\varphi =\frac{B_0\ell }{a}\stackrel{x^{\prime }+\ell }{\underset{x^{\prime }}{\int }}xdx$

$\Rightarrow \varphi =\frac{B_0\ell }{2a}\left[\right(x^{\prime }+\ell {)}^2-\left(x^{\prime }{)}^2\right]$

$\epsilon =\frac{d\varphi }{dt}=\frac{B_0\ell }{2a}\left[2\right(x^{\prime }+\ell )V_0-2x^{\prime }{V}_0]$

$\epsilon =\frac{B_0{\ell }^2{V}_0}{a}=3volts$