Q. A wooden stick of length 3*l *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} $