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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.
Solution
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.
Fm = 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]`
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