Question

A charged particle of charge e and mass m is moving in an electric field E and magnetic field B. Construct dimensionless quantities and quantities of dimension [T]^–1.

Answer 1

If a charged particle is moving in electric and magnetic field, we cannot construct any dimensionless quantity with these physical quantities.

For a charged particle moving perpendicular to the magnetic field, the magnetic Lorentz forces provide the necessary centripetal force for revolution.

F_m = qvB sin 90° = qvB  ......(1)

We know that centripetal force = `(mv^2)/R` ......(2)

By equation (1) and (2)

`(mv^2)/R = qvB`

`v/R = (qB)/m`

∵ v = ωR and q = e

Angular velocity

ω = `v/R = (eB)/m`

Dimensional formula for angular velocity ω

ω = `[(eB)/m] = [v/R] = [T^-1]`