Question

A uniform magnetic field of magnitude 0.165 T is directed along the positive x axis. A positron moving at a speed of 4.95 Times 10^6 m/s enters the field along a direction that makes an angle of theta = 85.0 degree with the x axis (see figure below). The motion of the particle is expected to be a helix. Calculate the pitch p of the trajectory as defined in figure. Calculate the radius r of the trajectory as defined in figure.

Answer 1

The first thing to realize is that a helix is basically a curve described by circular motion in one plane, in this case the y − z plane, and linear motion along the perpendicular direction, in this case the x axis. A helix of circular radius a and pitch p can be described parametrically by

x(t) = pt / 2π

y(t) = a cost

z(t) = a sin t

As we can see, the motion in the y − z plane obeys y 2 + z 2 =a 2 , describing a circle of radius a, and along the x axis we just have constant velocity motion. Since the x, y, and z motions are uncoupled things are in fact pretty simple

The circular motion comes from the component of the velocity perpendicular to the magnetic field, the component of velocity lying in the y − z plane, which we will call v⊥. The pitch is just how far forward along the x axis the particle moves in one period of circular motion T. Thus, if the velocity along the x axis is vx

p = vxT = (v cos 85 ) T

We have already discovered that the period and radius of circular motion for a particle in a magnetic field does not depend on the particle’s velocity, it only matters that there is always a velocity component perpendicular to the magnetic field

T = 2πm / qB,

r = mv⊥ / qB

p = 2πmv / Bq cos 85

where

m = 9.109 x 10^-31 kg

q = 1.602 x 10^-19 C

B = 0.165 T

v = 4.95 x 10^6

q = [(2 x 3.14 x 9.109 x 10^-31 x 4.95 x 10^6) / (1.602 x 10^-19 x 0.165)]cos85

q = 0.9336 x 10^-4 m

b)

r = [mv / qB] sin 85

r = [(9.109 x 10^-31 x 4.95 x 10^6) / (1.602 x 10^-19 x 0.165)]sin(85)

r = 1.70 x 10^-4 m