Question
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
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
A.
$$\frac{{2B\omega {\ell ^2}}}{2}$$
B.
$$\frac{{3B\omega {\ell ^2}}}{2}$$
C.
$$\frac{{4B\omega {\ell ^2}}}{2}$$
D.
$$\frac{{5B\omega {\ell ^2}}}{2}$$
Answer :
$$\frac{{5B\omega {\ell ^2}}}{2}$$
Solution :
Here, induced e.m.f
$$\eqalign{ & e = \int\limits_{2\ell }^{3\ell } {\left( {\omega x} \right)} Bdx = B\omega \frac{{\left[ {{{\left( {3\ell } \right)}^2} - {{\left( {2\ell } \right)}^2}} \right]}}{2} \cr & = \frac{{5B{\ell ^2}\omega }}{2} \cr} $$
Here, induced e.m.f
$$\eqalign{ & e = \int\limits_{2\ell }^{3\ell } {\left( {\omega x} \right)} Bdx = B\omega \frac{{\left[ {{{\left( {3\ell } \right)}^2} - {{\left( {2\ell } \right)}^2}} \right]}}{2} \cr & = \frac{{5B{\ell ^2}\omega }}{2} \cr} $$
