Q. 59

# 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?

Answer :

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) = μ_{0}ni,

where

μ_{0} = 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 = mv^{2}/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 = **qμ _{0}nir/2m** (Ans)

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