With a brief history of electromagnetism, and a general understanding of what
conditions give rise to a magnetic field, we may now precisely define the
magnetic field.

Magnetic Field Acting on a Charge

When we defined the electric field, we first established the electric charge,
and related the interaction of electric charges through Coulomb's Law.
Unfortunately we cannot do the same for magnetic fields, because magnetic
charges do not exist. Whereas electric fields originate from a single point
charge, magnetic fields come from a wide variety of sources: currents in wires
of varying shapes or forms, permanent magnets, etc. Instead of beginning with a
description of the field created by each of these examples, we must define the
magnetic field in terms of the force exerted by the field on a moving point
charge.

Consider a point charge q moving with a velocity v that is perpendicular to the
direction of the magnetic field, as shown below.

In this very simple case, the force felt by the positive point charge has
magnitude

F =

where B is the magnitude of the magnetic field, and c is the speed of light.
The force points in the positive z direction, as shown in the figure. Because
we are now working in three dimensions, it is often difficult to determine the
direction of this force. The easiest way to do this is to use your hands, as we
will explain.

First Right Hand Rule

Take your right hand (it is important not to use the left one), and stick
your thumb, your index finger and your middle finger in mutually perpendicular
directions. Each one of these fingers represents a vector quantity: the thumb
points in the direction of the velocity of the positively charged particle, the
index finger points in the direction of the magnetic field, and the middle
finger points in the direction of the force felt by the moving charge. Try it
out on the above figure: point your thumb in the negative x direction and your
index finger in the negative y direction. Hopefully you will find that your
middle finger points in the positive z direction, which is exactly the direction
of the force. This is known as the first right hand rule.

Magnetic Force when Moving Charges are not Perpendicular

We disucussed the special case in which the moving charge moves perpendicular to
the magnetic field. This perfectly perpendicular situation is uncommon. In
more normal circumstances the magnetic force is proportional to the component of
the velocity that acts in the perpendicular direction. If a charge moves with a
velocity at an angle θ to the magnetic field, the force on that particle
is defined as:

F =

If you are familiar with vector calculus, you will notice that this can be
simplified in terms of cross products:

forceequation*

=

This last equation is the most complete; the cross product of two vectors is
always perpendicular to both vectors, providing the correct direction for the
direction of our force.