Question

A tightly-wound, long solenoid has n turns per unit length, a radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in a direction perpendicular to the axis. What can be the maximum speed for which the particle does not strike the solenoid?

Answer 1

Given:
Magnitude of current in the solenoid = i
Number of turns per unit length = n
When a particle is projected perpendicular to the magnetic field, it describes a circular path.
And for the particle (projected from a point on the axis in a direction perpendicular to the axis) to not strike the solenoid, the maximum radius of that circular path should be r/2.
 
∴ Radius of the circle = \[\frac{r}{2}\]

O is a point on the axis of the solenoid . The magnetic  field of the solenoid is along the axis, Which is perpendicular to the plane of paper . 

We know,
Centripetal force = Magnetic force

\[\frac{m V^2}{r} = qVB\]
\[ \Rightarrow V = \frac{qBr}{m} = \frac{q \mu_0 nir}{2m}\]