Consider a particle of mass m and charge q moving in the -plane under the influence of a uniform magnetic field pointing in the direction. We have to prove that the particle moves in a circular path by solving Newton’s equations of motion analytically.

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Force on a particle of charge q, mass m, moving in magnetic field is

.

Let the position vector of the particle be .

We use the Newton’s second law for writing the equation of motion. It is

It is given that the magnetic field is in the direction. That is

As the particle is moving in the -plane,

Substituting this in the equation of motion, we have

Without loss of generality, we select the solution of this equation as

Therefore,

Note that

That is the charged particle is moving with constant speed and describes a circular motion. Let the radius of the circular orbit be r. Integrating and equations, we can write

By plotting and , we note if is positive, the particle will move in the clockwise direction.