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(B) The magnetic field inside a long solenoid is uniform and directed along its axis,given by $B = \mu_{0} n I$. When a particle of charge $q$ and mas

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$\frac{\mu_{0} n I q r}{2 m}$

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(B) The magnetic field inside a long solenoid is uniform and directed along its axis,given by $B = \mu_{0} n I$. When a particle of charge $q$ and mass $m$ is projected perpendicular to the axis,it experiences a Lorentz force $F = q(v 	imes B)$. Since $v \perp B$,the force is $F = qvB$,which acts as the centripetal force. The particle moves in a circular path of radius $R_{c} = \frac{mv}{qB}$. For the particle not to strike the solenoid wall,the diameter of its circular path must be less than or equal to the radius of the solenoid $r$. Thus,$2R_{c} \leq r$,which implies $R_{c} \leq \frac{r}{2}$. Substituting $R_{c} = \frac{mv}{qB}$,we get $\frac{mv}{qB} \leq \frac{r}{2}$. Solving for $v$,we get $v \leq \frac{qBr}{2m}$. Substituting $B = \mu_{0} n I$,the maximum speed is $v_{max} = \frac{\mu_{0} n I q r}{2m}$.

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