question_answer

A metal rod of length  moving with an angular velocity  and velocity of its centre is v. Find potential difference between points A and B at the instant shown in figure. A uniform magnetic field of strength B exist perpendicular to plane of paper:

A) \[B\,v\,\ell \]

B) \[Bv\ell +\frac{1}{2}B\omega \,{{\ell }^{2}}\]

C) \[B\omega \,\ell -\frac{1}{2}B\omega \,{{\ell }^{2}}\]

D) \[Bv\,\ell +B\,\omega \,\,{{\left( \frac{\ell }{2} \right)}^{2}}\]

Correct Answer: A

Solution :

Point P is at instantaneous rest,

\[{{\varepsilon }_{1}}=\,\,|{{v}_{P}}-{{v}_{A}}|\,\,=\frac{1}{2}B\omega \,\,{{\left( \frac{\ell }{2}+\frac{v}{\omega } \right)}^{2}}\]

\[{{\varepsilon }_{2}}=\,\,|{{v}_{P}}-{{v}_{B}}|\,\,=\frac{1}{2}B\omega \,\,{{\left( \frac{\ell }{2}-\frac{v}{\omega } \right)}^{2}}\]

\[|{{v}_{A}}-{{v}_{B}}|\,\,={{\varepsilon }_{1}}-{{\varepsilon }_{2}}\]

\[|{{v}_{A}}-{{v}_{B}}|\,\,=B\ell v\]