The force exerted on a charge in an electric field is given by
F = qE.

What is the equation for the force exerted on a charge in a magnetic field?

Case 1 - The charge is stationary in the magnetic field. What can you conclude?

Case 2 - The charge is moving parallel to the field. What can you conclude?

Case 3 - Three objects of equal mass, one neutral, one negative, and one positive, are all initially moving with the same velocity perpendicular to the field, which is directed out of the screen. What can you conclude?

Case 4 - The same as case 3 except the magnitude of the magnetic field is doubled. What can you conclude?

Case 5 - Three positive charges +q, +2q, and +3q are initially moving perpendicular to the field with the same velocity. What can you conclude?

Case 6 - Here we have equal charges initially moving perpendicular to the field. The initial velocities are v, 2v, and 3v, respectively.

The charges follow circular paths at constant speed. We can apply uniform circular motion ideas:
F = mv²/r

This gives r = mv²/F

Note that we observe that r is proportional to v.

Some of your conclusions should include:

• There is no force on a stationary charge, or on a charge moving parallel to the field.
• The direction of the force experienced by a positive charge is opposite to that experienced by a negative charge if the charges are moving in the same direction.
• The force is proportional to q
• The force is proportional to B
• The force is proportional to v

We can satisfy all these observations with the following force equation:

F = qv × B

The magnitude of the force is F = qvB sin(θ).

The direction of the force is given by the right-hand rule.

The force exerted on a charge moving in a magnetic field is always perpendicular to both the velocity and the field. Any force perpendicular to the velocity can not change the speed (or the kinetic energy). All it can do is make the charge change direction.