Motion In A Crossed Electric And Magnetic Fields

Motion in a crossed electric and magnetic fields: The force on the charged particle in the presence of both electric and magnetic fields is given by

Let the electric and magnetic fields be at right angle to each other, so that,

If the particle is initially at rest no magnetic force acts on the particle. As the electric field exerts a force on the particle, it acquires a velocity in the direction of . The magnetic force now acts sidewise on the particle.
For a quantitative analysis of the motion, let be taken along the x-direction and along z-direction. As there is no component of the force along the z-direction, the velocity of the particle remains zero in this direction. The motion, therefore, takes place in x-y plane. The equations of motion are

As in the earlier case, we can solve the equations by differentiating one of the equations and substituting the other,

which, as before, has the solution

with . Substituting this solution into the equation for , we get

Since , the constant , so that

The constant $A$may be determined by substituting the solutions in eqn. (1) which gives

Since the equation above is valid for all times, the constant terms on the right must cancel, which gives . Thus we have