Question

The magnetic field in a region is given by $B=3r$, where $r$ is the distance from the origin. The direction of the magnetic field is along the negative $\text{z-axis}$. A conducting rod of length $20\text{cm}$ lies as shown in the figure. If the rod is rotating with an angular velocity of $4\text{rad/s}$, find the emf induced between the ends of the rod.

• A. $1.6\times {10}^{-2}\text{V}$
• B. $3.2\times {10}^{-2}\text{V}$
• C. $6\times {10}^{-2}\text{V}$
• D. $4\times {10}^{-2}\text{V}$

Answer 1

The correct option is B $3.2\times {10}^{-2}\text{V}$

Consider an element of length $dr$ at a distance $r$.


Emf induced in this element is:

$dE=Bvl=B\left(\omega r\right)dr$

$dE=3r\times 4\times rdr=12r^{2}dr$

$\therefore$ Total emf developed across the ends of the rod is,

$E=\int dE={\int }_0^{0.2}12r^{2}dr$

$=12\times {\left[\frac{r^3}{3}\right]}_0^{0.2}=4\times (0.2{)}^3$

$\therefore E=3.2\times {10}^{-2}\text{V}$

Hence, $\left(\text{B}\right)$ is the correct answer.