A proton moving with a speed of 1.50 × 10 ^6 m/s enters a region containing a un

Question

A proton moving with a speed of 1.50 × 10 ^6 m/s enters a region containing a uniform magnetic ?eld of magnitude 0.500 T pointing in the -z direction. The velocity vector is at an angle of 60? with respect to the +z axis.

(a) Describe the trajectory of the proton in qualitative terms. Justify your answer in terms of the force on the proton.
(b) Calculate the radius of the trajectory projected onto a plane perpendicular
to the ?eld (i.e., in the x-y plane).
(c) Calculate the period, the frequency, and the angular frequency of the
motion in the x-y plane.
(d) What is the pitch of the motion (the distance traveled along the z axis in one
period of the circular motion in the x-y plane)?

Explanation / Answer

(a) The trajectory is a helix whose axis is parallel to the magnetic field direction. The motion can be decomposed into uniform circular motion in the xy plane perpendicular to the magnetic field direction and uniform velocity opposite the direction of the magnetic field.  The magnetic force acting on the proton is perpendicular to the component of the proton velocity perpendicular to the magnetic field.

(b) In the xy plane, the protons execute uniform circular motion (UCM), that is constant speed in a circular path. The component of the velocity in this plane (the plane perpendicular to the magnetic field) is given by

where θ is the angle the velocity makes with the +z axis.  Since the motion in this plane is UCM, the magnetic force must equal the product of the mass and the centripetal acceleration.  The magnitude of the  magnetic force is given by

,

where B is the magnitude of the magnetic field, thus

,

where m=mass of the proton and r=radius of the circular path in the xy plane.  From this equation, we obtain the radius asked for in the problem.

(c) Once we have the radius, we can calculate the period T by setting the distance to go around the circular path once equal to the perpendicular velocity component times the period:

.

From T we get the linear frequency, as

,

and the angular frequency from

.

(d) The pitch is the distance the proton travels along the magnetic field direction during one circuit of the circular motion:

.

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