Question
A wooden stick of length $$3\ell $$ is rotated about an end with constant angular velocity $$\omega $$ in a uniform magnetic field $$B$$ perpendicular to the plane of motion. If the upper one third of its length is coated with copper, the potential difference across the whole length of the stick is
A wooden stick of length $$3\ell $$ is rotated about an end with constant angular velocity $$\omega $$ in a uniform magnetic field $$B$$ perpendicular to the plane of motion. If the upper one third of its length is coated with copper, the potential difference across the whole length of the stick is
A.
$$\frac{{9B\omega {\ell ^2}}}{2}$$
B.
$$\frac{{4B\omega {\ell ^2}}}{2}$$
C.
$$\frac{{5B\omega {\ell ^2}}}{2}$$
D.
$$\frac{{B\omega {\ell ^2}}}{2}$$
Answer :
$$\frac{{5B\omega {\ell ^2}}}{2}$$
Solution :
$${\text{For}}\,A:{e_A} = \frac{1}{2}B\omega {\left( {3\ell } \right)^2}\,{\text{and}}\,{e_B} = \frac{1}{2}B\left( { - \omega } \right){\left( {2\ell } \right)^2}$$
Resultant induced emf will be
$$e = {e_A} + {e_B} = \frac{1}{2}B\omega {\ell ^2}\left( {9 - 4} \right)\,{\text{or}}\,e = \frac{5}{2}B\omega {\ell ^2}$$
$${\text{For}}\,A:{e_A} = \frac{1}{2}B\omega {\left( {3\ell } \right)^2}\,{\text{and}}\,{e_B} = \frac{1}{2}B\left( { - \omega } \right){\left( {2\ell } \right)^2}$$
Resultant induced emf will be
$$e = {e_A} + {e_B} = \frac{1}{2}B\omega {\ell ^2}\left( {9 - 4} \right)\,{\text{or}}\,e = \frac{5}{2}B\omega {\ell ^2}$$
