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 = v₀ $\overrightarrow{i}$, find the emf induced between the ends of the rod.

Answer 1

Magnetic field in the given region, $\overrightarrow{B}=\frac{B_0}{L}y\hat{k}$
Length of the rod on the y-axis = L
Velocity of the rod, v = v₀$\hat{i}$
We will consider a small element of length dy on the rod.
Now,
Emf induced in the element:
de = Bvdy

$$\begin{gathered} \Rightarrow de=\frac{B_0}{L}y\times v_0\times dy \\ =\frac{B_0{v}_0}{L}y\mathrm{d}y \\ \mathrm{And}, \\ e=\frac{B_0{v}_0}{L}\underset{0}{\overset{\mathrm{L}}{\int }}ydy \\ =\frac{B_0{v}_0}{L}{\left[\frac{y^2}{2}\right]}_0^L \\ =\frac{B_0{v}_0}{L}\frac{L^2}{2} \\ =\frac{1}{2}{B}_0{v}_{0}L \end{gathered}$$