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?

Given:


Number of turns per unit length = n


Current = i


Radius = r


Charge of particle = q


Mass = m


Formula used:


Magnetic field inside a solenoid(B) = μ0ni,


where


μ0 = magnetic permeability of vacuum = 4π x 10-7 T m A-1,


n = number of turns per unit length,


i = current carried by the wire


Now, when a particle is projected perpendicular to a magnetic field, it describes a circle. Now, for the particle to not strike to solenoid, the required radius is r/2.


Since it is moving in a circular path, centripetal acceleration = mv2/r,


Where


m = mass of particle,


v = velocity,


r = radius


Force due to the magnetic field = qvB,


Where


q = charge of particle,


v = velocity,


B = magnetic field.


Here, r => r/2.


The centripetal and magnetic forces balance each other.


Hence, using the given data:


=> v = 0nir/2m (Ans)


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