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