Deduce the expression for the frequency of revolution of a charged particle in a magnetic field and show that this is independent of the velocity or energy of the particle.

Explanation

To deduce the expression for the frequency of revolution of a charged particle in a magnetic field and show that it is independent of the velocity or energy of the particle, we can follow these steps:

### Step 1: Understand the Forces Acting on the Charged Particle
When a charged particle with charge \( q \) moves in a magnetic field \( \mathbf{B} \), it experiences a magnetic force given by:
\[ \mathbf{F}_m = q \mathbf{v} \times \mathbf{B} \]
For a particle moving perpendicular to the magnetic field, the magnitude of the magnetic force can be simplified to:
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