Question

The magnetic field in a region is given by $\overrightarrow{B}=\overrightarrow{k}\frac{B_0}{L}y$ where L is a fixed length. A conducting rod of length L lies along the Y -axis between the origin and the point (0,L,0). IF the rod moves with a velocity $v=v_0\overrightarrow{i}$, find the emf induced between the ends of the rod.

Answer 1

$\overrightarrow{B}=\frac{B_0}{L}y\hat{k}$

L =Length of rod on y axis

$V=V_0\hat{i}$

Considering a small length dy to the rod

dE=BV dy

$\Rightarrow dE=\frac{B_0}{L}y\times V_0\times dy$

$\Rightarrow dE=\frac{B_0{V}_0}{L}ydy$

$E=\frac{B_0{V}_0}{L}{\int }_0^{L}ydy$

$=\frac{B_0{V}_0}{L}={\left[\frac{y^2}{2}\right]}_0^L$

$=\frac{B_0{V}_0}{L}\frac{L^2}{2}$

$=\frac{1}{2}{B}_0{V}_{0}L$