Question

A metallic rod of length ${}^{\prime }{\ell }^{\prime }$ is tied to a string of length $2\ell$ and made to rotate with angular speed $\omega$ on a horizontal table with one end of the string fixed. If there is a vertical magnetic field ${}^{\prime }{B}^{\prime }$ in the region, the e.m.f. induced across the ends of the rod is:

• A. $\frac{3B\omega {\ell }^2}{2}$
• B. $\frac{4B\omega {\ell }^2}{2}$
• C. $\frac{5B\omega {\ell }^2}{2}$
• D. $\frac{2B\omega {\ell }^2}{2}$

Answer 1

The correct option is C $\frac{5B\omega {\ell }^2}{2}$

$\begin{array}{c}V=\int \left(\overrightarrow{N}\times \overrightarrow{B}\right).\overrightarrow{dt}\\ V={\int }_{2l}^{3l}xWB.dx={\frac{x^2}{2}WB|}_{2l}^{3l}\\ =\frac{9l^{2}WB}{2}-\frac{4l^{2}WB}{2}=\frac{5l^{2}WB}{2}\end{array}$