question_answer

A metallic rod of length \['\ell '\] is tied to a string of length \[2\ell \] and made to rotate with angular speed co 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

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 :

Here, induced e.m.f.   \[e=\int\limits_{2\ell }^{3\ell }{(\omega x)Bdx=B\omega \frac{[{{(3\ell )}^{2}}-{{(2\ell )}^{2}}]}{2}}\] \[=\frac{5B{{\ell }^{2}}\omega }{2}\]