Question

Problem 2: A particle with charge q is emitted from the origin with momentum p directed at an angle ? to a uniform magnetic field B which lies in the z-direction. i) What is the position of the particle as a function of time? ii) At what point does the particle next intersect the z-axis?

Answer 1

At time t = 0, the charge is at the origin with an initial velocity,  

The equation of motion for the particle of mass m and charge q in a uniform magnetic field in the z-direction is

The equation of the motion in x, y & z direction are as follow:-

(b)

At t=0 particle is at origin.The particle crosses the z axis if x(t)=0 and y(t)=0.

From the motion equation , we can see that x(t) = 0 when

with n as an integer. Similarly y(t) = 0 when   where n' is an different integer.

First time after t=0 both x(t) & y(t) become zero is when n=2 and n' =1.

Thus, . This is equivalent to one gyro-period.

Hence, at every time which is a multiple of the gyro-period, the charged particles intersects the z-axis.