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

• A. \(\frac{\mu_onlqr}{m}\)
• B. \(\frac{\mu_onlqr}{2m}\)
• C. \(\frac{\mu_onlqr}{4m}\)
• D. \(\frac{\mu_onlqr}{8m}\)

Answer 1

The Correct Option is B

\[F_c = \frac{mv^2}{r}\]

For the particle to undergo circular motion, these two forces should be equal:

\[F_B = F_c\]

\[\Rightarrow qvB = \frac{mv^2}{r}\]

\[\Rightarrow v = \frac{qrB}{2m}\]

But we also know the magnetic field due to a solenoid with 'n' turns per unit length as:

\[B = \mu_0ni\]

Substituting this in the equation for velocity gives:

\[\therefore v = \frac{q\mu_0nir}{2m}\]

The velocity at which the particle moves should be less than or equal to \(\frac{\mu_0gnir}{2m}\).