Question

The magnetic field in a region is given by $\overrightarrow{B}=B_0\left(\frac{x}{a}\right)\hat{k}$. A square loop of side $d$ is placed with it's edges on the x and y axis. The loop is moved with a constant velocity $\overrightarrow{v}=v_0\hat{i}$. The emf induced in the loop is :

• A. $B_0{v}_{0}d$
• B. $\frac{B_0{v}_0{d}^2}{2a}$
• C. $\frac{B_0{v}_0{d}^3}{a^2}$
• D. $\frac{B_0{v}_0{d}^2}{a}$

Answer 1

The correct option is D $\frac{B_0{v}_0{d}^2}{a}$

$\overrightarrow{B}=B_o\frac{x}{a}\hat{K}$

$\overrightarrow{V}=V_o\hat{i}=\frac{dx}{dt}\hat{i}$

consider a strips at x distance from origin
$d\Phi =(d\times dx)\left(B_o\frac{x}{a}\right)$

$\Phi ={\int }_x^{d+x}\frac{d{B}_o}{a}xdx$

$\Phi =\frac{B_{o}d}{2a}\left[\right(d+x{)}^2-x^2]$

$\Phi =\frac{B_{o}d}{2a}[d^2+2dx]$

emf induced $=\frac{d\Phi }{dt}=\frac{B_{o}d}{2a}\times 2d\times \frac{dx}{dt}$

$=\frac{B_o{V}_o{d}^2}{a}$