Q. A tightly-wound long solenoid has n turns per unit length, radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in the direction perpendicular to the axis. The maximum speed for which particle does not strike the solenoid will be
(a) $\displaystyle \frac {\mu_0 q r n i}{2m}$
(b) $ \displaystyle \frac {\mu_0 q r n i}{m}$
(c) $ \displaystyle \frac {2\mu_0 q r n i}{3m}$
(d) None of these
Ans: (a)
Sol: Magnetic field inside the solenoid is B = μ0 n i , along the axis of solenoid
Since velocity is Perpendicular to B , hence particle moves in a circle of radius
$ \displaystyle R = \frac{m v}{q B} = \frac{m v}{q \mu_0 n i} $
If particle does not strike solenoid
R ≤ r/2
$ \displaystyle \frac{m v}{q \mu_0 n i} \le \frac{r}{2} $
$ \displaystyle v \le \frac{\mu_0 n i q r}{2 m} $
$\displaystyle v_{max} = \frac{\mu_0 n i q r}{2 m} $