JEE MAIN - Physics (2013 (Offline) - No. 10)
A metallic rod of length $$'\ell '$$ is tied to a string of length $$2$$$$\ell $$ and made to rotate with angular speed $$w$$ 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
_en_10_1.png)
$${{2B\omega \ell } \over 2}$$
$${{3B\omega \ell } \over 2}$$
$${{4B\omega {\ell ^2}} \over 2}$$
$${{5B\omega {\ell ^2}} \over 2}$$
Explanation
Here, induced $$e.m.f.$$
_en_10_2.png)
$$e = \int\limits_{2\ell }^{3\ell } {\left( {\omega x} \right)Bdx = B\omega } {{\left[ {{{\left( {3\ell } \right)}^2} - {{\left( {2\ell } \right)}^2}} \right]} \over 2}$$
$$ = {{5B{\ell ^2}\omega } \over 2}$$
$$e = \int\limits_{2\ell }^{3\ell } {\left( {\omega x} \right)Bdx = B\omega } {{\left[ {{{\left( {3\ell } \right)}^2} - {{\left( {2\ell } \right)}^2}} \right]} \over 2}$$
$$ = {{5B{\ell ^2}\omega } \over 2}$$
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