Question

Obtain the expression for the cyclotron frequency.

Answer 1

When a charged particle having charge q moves inside a magnetic field `vecB`with velocity v, it experiences a force `vecF=q(vecvxxvecB)`.

When `vecv` is perpendicular to`vecB`, the force `vecF` on the charged particle acts as the centripetal force and makes it move along a circular path.
​Let m be the mass of charged particle and r be the radius of the circular path.

Then `q(vecv xx vecB )=(mv^2)/r`

∵ v and B are at right angles

`∴ qvB=(mv^2)/r`

`⇒r=(mv)/(Bq)`

Time period of the circular motion of a charged particle is given by

`T=(2pir)/v`

`T=(2pi)/v (mv)/(Bq)`

`therefore T=(2pim)/(Bq)`

Angular frequency,`omega=(2pi)/T`

`therefore omega=(Bq)/m`

This is often called cyclotron frequency.