Four factors determine the amount of force exerted by a magnetic field on a moving
particle. They are the particle’s charge and speed, the strength of the magnetic field and
the angle of intersection between the particle’s velocity and the magnetic field. The
force is greatest when these two vectors are perpendicular, and zero when they are
parallel.
When a charged particle is surrounded by an external electric field, the electric force on
the charge is exerted along the field lines. The electric field exerts a force on the charge
whether it is moving or stationary. In contrast, magnetic fields only exert a force on
moving charged particles, pushing them neither in the direction of the particle’s motion
nor along the lines of the field. The force exerted on a moving charge by a magnetic
field is perpendicular to both the particle’s velocity and the direction of the field.
You see this illustrated to the right, and you also experienced it when you used the two
simulations in the introduction to this chapter. In Concept 1, the same phenomenon is
shown from two different vantage points.
In both illustrations, a positive charge is moving through a magnetic field. In the view
labeled “side view,” the magnetic field points away from you. This is depicted with ×’s,
which represent the field lines viewed from behind. This view is used to best show the
direction of the force: It is perpendicular to both the field and the velocity vectors.
In the view labeled “front view,” your viewpoint has been rotated 90° so the field
appears parallel to the screen. The force vector in the front view points toward you and
is represented by a dot. You are looking at “the business end” of the vector’s arrow.
A right-hand rule is used to determine the direction of the force. The front view makes it
easier to see, so we use that in Concept 2. To apply the right-hand rule, the fingers of a
flat hand start out pointing in the same direction as the velocity vector of the charge.
Then they curl, so that the fingers point in the same direction as the magnetic field. The
fingers wrap from the velocity vector to the magnetic field vector.
For a positive charge, the thumb points in the direction of the force exerted on the
moving charge. The thumb points opposite to the direction of the force for a negative
charge. In other words, with a negative charge, you apply the same rule, but reverse
the results.
The equation to determine the force is shown to the right. The force equals the charge
times the cross product of the velocity and magnetic field vectors. The cross product is
calculated with the sine, as shown to the right.
To determine the amount of force, multiply the absolute value of the charge (its positive
value), the charge’s speed, the field strength, and the sine of the angle between the
velocity and field vectors. This angle is shown in the illustration for the equation. When
calculating the force magnitude, you use the smaller, positive angle between the
velocity and magnetic field vectors. For instance, in the Equation 1 diagram the angle is
90°, not í90° or 270°.
As mentioned earlier, the unit for magnetic field strength is the tesla. One tesla equals
one newton·second per coulomb·meter, or N·s/C·m. In other words, one coulomb of
charge traveling at one meter per second through a magnetic field having a strength of
one tesla experiences one newton of force. A tesla is a rather large unit (remember that
one coulomb is a lot of charge), so the smaller unit gauss (G) is fairly common. Ten
thousand gauss equals one tesla. As mentioned earlier, the Earth’s magnetic field is
about 5×10í^5 T, which equals 0.5 G.
Magnetic field, charged particle
Magnetic field exerts force on moving
charge
·Force perpendicular to velocity, fieldRight-hand rule
Determines direction of force
·Fingers curl from v to B
·Thumb shows force on positive charge