Q. A wooden stick of length 3l is rotated about an end with constant angular velocity w 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) $\displaystyle \frac {9 B \omega l^2}{2} $
(b) $ \displaystyle \frac {4 B \omega l^2}{2} $
(c) $\displaystyle \frac {5 B \omega l^2}{2} $
(d) $ \displaystyle \frac { B \omega l^2}{2} $
Ans: (c)
Sol: potential difference across the whole length of the stick is
$ \displaystyle = \int_{2l}^{3l} B(dx)\omega x $
$ \displaystyle = B \omega \int_{2l}^{3l} x dx $
$\displaystyle = B \omega [\frac{x^2}{2}]_{2l}^{3l} $
$ \displaystyle = B \omega [\frac{(3l)^2}{2}- \frac{(2l)^2}{2}] $
$ \displaystyle = B \omega [\frac{5l^2}{2}] $
$\displaystyle = \frac {5 B \omega l^2}{2} $