question_answer

A metallic rod of length\[\ell \] 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 'B' in the region, the e.m.f. induced across the ends of the rod is:     AIEEE Solevd Paper-2013

A) \[\frac{2B\omega {{\ell }^{2}}}{2}\]                        

B) \[\frac{3B\omega {{\ell }^{2}}}{2}\]

C) \[\frac{4B\omega {{\ell }^{2}}}{2}\]        

D) \[\frac{5B\omega {{\ell }^{2}}}{2}\]

Correct Answer: D

Solution :

\[\int{\left( \vec{v}\times \vec{B} \right)}.\overrightarrow{d\ell }=\int\limits_{2\ell }^{3\ell }{x\omega Bdx}\] \[V=\frac{5B\omega {{\ell }^{2}}}{2}\]