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. The maximum speed of the particle for which the particle does not strike the solenoid is

Magnetic force on the charged particle is given as
\(F_{B}=q(v \times B)=q v B \sin \theta\)
Here, \(\theta=90^{\circ}\)
\(\Rightarrow F_{B}=q v B \sin 90^{\circ}=q v B\)
Since, the particle is projected perpendicular to the field, so its trajectory will be circular in nature, where necessary centripetal force \(=F_{B}\)
\(\Rightarrow \frac{m v^{2}}{r^{\prime}}=q v B\)
\(\frac{m v^{2}}{\left(\frac{r}{2}\right)}=q v B \quad\left(\because r^{\prime}=\frac{r}{2}\right)\)
\(\Rightarrow v=\frac{q B r}{2 m}...(i)\)
Magnetic field inside the solenoid is given as,
\(B=\mu_{0} n I\)
Substituting the value of \(B\) in Eq. (i), we get
\(v=\frac{\mu_{0} n I q r}{2 m}\)