


Magnetic Force on Charges
The questions on magnetism that you’ll find on SAT II
Physics will deal for the most part with the reciprocal relationship
between magnetic fields and moving charges. Generally, these questions
will expect you to predict the motion of a charge through a magnetic
field, or to calculate the magnitude of the magnetic force or magnetic
field strength necessary to move a charge in a certain manner.
Calculating Magnetic Force
A magnetic field exerts a force on a moving charge. Given
a magnetic field, B,
and a charge, q, moving with velocity, v,
the force, F, on the
charge is:
Magnetic field strength is measured in teslas (T),
where 1 T = 1 N/A · m.
You’ll notice that the force on a moving particle is calculated
as a cross product of the particle’s velocity and the magnetic field’s
strength. You can determine the direction of the vector by using the righthand rule as
follows: point the fingers of your right hand in the direction of
the velocity vector and then curl them around to point in the direction
of the magnetic field vector. The direction in which your thumb
points gives you the direction of the vector.
However, though q is a scalar
quantity, it can affect the direction of the force vector. If q has
a negative value, then has a negative value,
and so the force vector will point in a direction opposite from
what the righthand rule might tell you.
You can calculate the magnitude of the magnetic force
without using the righthand rule, so long as you know the angle, , between the velocity vector and the magnetic
field vector:
The sin term is important, because
it lets us see very quickly that there is no force if a charge moves
parallel to a magnetic field, and that the greatest force occurs
when a charge moves perpendicular to the magnetic field.
Example

The cross product of is a vector of magnitude qvB sin = 3 N. Following the righthand
rule, point your fingers toward the top of the page, and then curl
them around so that they point into the page. You’ll find that your
thumb is pointing left, which is the direction of the vector. Because the value of q is
positive, the force acting on the particle will also be in the leftward
direction.
A Quick Note on Vectors Going In and Out of the
Page
The magnetic field lines illustrated in this example that
are going into the page are represented by circles with an “x” inscribed
in them. Vector lines pointing out of the page are represented by
circles with a dot in them. You can think about these symbols as
arrows appearing from in front or behind: from in front, you see
the conical tip of the arrow, and from behind you see the fletching
of the four feathers in an “x” shape.
Trajectory of Charges in a Magnetic Field
The direction of the force on a moving charge
depends on the direction of its velocity. As its velocity changes,
so will its direction. The magnitude of the velocity will
not change, but charged particles moving in a magnetic
field experience nonlinear trajectories.
When the Velocity Vector and Magnetic Field Lines
Are Perpendicular
In the example above, we saw that a force of 3 N
would pull the charged particle to the left. However, as soon as
the particle begins to move, the velocity vector changes, and so
must the force acting on the particle. As long as the particle’s
velocity vector is at a right angle to the magnetic field lines,
the force vector will be at right angles to both the velocity vector and
the magnetic field. As we saw in the chapter on circular motion
and gravitation, a force that always acts perpendicular to the velocity
of an object causes that object to move in circular motion.
Because the velocity vector and the magnetic field lines
are at right angles to one another, the magnitude of the magnetic
force is F = qvB. Furthermore, because
the magnetic force pulls the particle in a circular path, it is
a centripetal force that fits the equation F = mv^{2}/r. Combining
these two equations, we can solve for r to
determine the radius of the circle of the charged particle’s orbit:
When the Velocity Vector and Magnetic Field Lines
Are Parallel
The magnetic force acting on a moving charged particle
is the cross product of the velocity vector and the magnetic field
vector, so when these two vectors are parallel, the magnetic force
acting on them is zero.
When the Velocity Vector and Magnetic Field Lines
Are Neither Perpendicular nor Parallel
The easiest way to deal with a velocity vector that is
neither parallel nor perpendicular to a magnetic field is to break
it into components that are perpendicular and parallel to the magnetic
field.
The xcomponent of the velocity vector
illustrated above will move with circular motion. Applying the righthand
rule, we find that the force will be directed downward into the page
if the particle has a positive charge. The ycomponent
of the velocity vector will experience no magnetic force at all,
because it is moving parallel to the magnetic field lines. As a result,
the charged particle will move in a helix pattern, spiraling around
while also moving up toward the top of the page. Its trajectory
will look something like this:
If the particle has a positive charge it will move in
a counterclockwise direction, and if it has a negative charge it
will move in a clockwise direction.
Example

We know the velocity, mass, charge, and radius of the
orbit of the particle. These four quantities are related to magnetic
field strength, B, in the equation r
= mv/qB. By rearranging this equation, we can solve
for B:
Now we just need to determine the direction of the magnetic
field. To find the direction, apply the righthand rule in reverse:
point your thumb in the direction of the force—toward the center
of the circle—and then stretch your fingers in the direction of
the velocity. When you curl your fingers around, they will point
out of the page. However, because the particle has a negative charge,
the magnetic field has the opposite direction—into the page.
Magnetic Fields and Electric Fields Overlapping
There’s no reason why a magnetic field and an electric
field can’t operate in the same place. Both will exert a force on
a moving charge. Figuring out the total force exerted on the charge
is pretty straightforward: you simply add the force exerted by the
magnetic field to the force exerted by the electric field. Let’s
look at an example.
Example

Answering this question is a matter of calculating the
force exerted by the magnetic field and the force exerted by the
electric field, and then adding them together. The force exerted by
the magnetic field is:
Using the righthandrule, we find that this force is
directed to the left. The force exerted by the electric field is:
This force is directed to the right. In sum, we have one
force of 6 N pushing the particle to the left and one
force of 6 N pushing the particle to the right. The
net force on the particle is zero, so it continues toward
the top of the page with a constant velocity of 10 m/s.
