Introductory Physics Volume Two

84 Magnetic Fields 4.3
For a magnetic force, ⃗v and ⃗ F are prependicular so that ⃗ F ·⃗v = 0. Thus
dW = 0 for a magnetic force.
Theorem: Work by Magnetic Force
The magnetic force does zero work. Thus the magnetic force can
change the direction but not the speed of a particle.
Because the magnetic force changes only the direction of a particle,
a magnetic field is useful for steering charged particles, once they are
already moving.
Consider a particle that is in a region with a uniform magnetic
field. Suppose that the particles initial velocity is perpendicular to the
magnetic field. Since the magnetic force is always perpendicular to
the direction of the magnetic field, the force will have no component
parallel to the field. So, since the particle started with zero velocity
parallel to the field it will continue to have zero velocity parallel to the
field. But this means that the velocity will always be perpendicular to
the magnetic field, so |⃗v × B| ⃗ = vB. Thus we know that the magnitude
of the magnetic force is
F ≡ | ⃗ F | = q|⃗v × ⃗ B| = qvB
In addition we know by the previous theorem that the speed of the
particle will be constant, v = v 0 . So that the magnitude of the force
is constant. In summary, we see that the magnitude of the magnetic
force on the particle is constant and perpendicular to the velocity, so
that the acceleration of the particle is constant and perpendicular to
the velocity. We have run into this situation before: In circular motion
the acceleration is constant and always perpendicular to the velocity.
We are lead by this observation to the following theorem.
Theorem: Circular Trajectories
If a particle is in a region of uniform magnetic field and entered
the region with a velocity perpendicular to the magnetic field, then
the particle will execute circular motion while in the region.
One can relate the radius and velocity of the motion to the field
strength and charge by employing Newton’s second law.
F = ma
−→ qvB = m v2
r
⊲ Problem 4.3
One can use the circular motion of a charge particle to determine it’s
mass if you already know it’s charge. Suppose that you send a particle