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.

#### Solution

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 = v_{0 }\[\hat i\]

We will consider a small element of length dy on the rod.

Now,

Emf induced in the element:-

de = Bvdy

\[\Rightarrow de = \frac{B_0}{L}y \times v_0 \times dy\]

\[ = \frac{B_0 v_0}{L}ydy\]

And,

\[e = \frac{B_0 v_0}{L} \int\limits_0^L ydy\]

\[ = \frac{B_0 v_0}{L} \left[ \frac{y^2}{2} \right]_0^L \]

\[ = \frac{B_0 v_0}{L}\frac{L^2}{2}\]

\[ = \frac{1}{2} B_0 v_0 L\]