Download Motion of a Point Charge in a Magnetic Field

Motion of a Point Charge in a Magnetic Field
We have already seen the basics of this material in our laboratory on the e/m of the
electron. There we learned that the magnetic force on a point charge moving with a velocity in a
G
G G
magnetic field is given by FB = qv × B . If the charge has its initial velocity perpendicular to the
magnetic field, and the magnetic field is uniform, then we can write that
v2
FB = qvB = ma = m ,
r
where we know that the charge will move in a circle, having the magnetic force directed toward
the center of the circle of radius r. Solving, we find that the radius of the circle is given by
mv
r=
.
qB
An additional feature, not seen in laboratory, is that since the speed is a constant going
around the circle, we can calculate the time to go around the circle, T, or the period of the
motion, as
2π r
T=
,
v
or, defining the angular velocity ω = v/r = qB/m (from above solving for r/v), we have
2π
m
T=
= 2π
= constant .
ω
qB
The time to travel around the circle once turns out to be independent of the size of the circle and
the speed. This ends up being very useful in some applications.
If there is an electric field present as well, then the net force on a charge q is given by the
most general equation, called the Lorentz force,
G
G
G G
F = qE + qv × B .