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 = qμ0nir/2m (Ans)