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 = μ_{0} 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} $