Physics 12: Magnetism: Magnetic Field and Force

A micro-course in magnetism: Magnetic field, and magnetic force acting on a moving charge and charge carrying wire, for high school students.

Magnetic Force

Similar to the electric charge that exerts attracting or repulsing force to the other charges, a magnet exerts magnetic attracting or repulsing forces to the other magnetic bodies. There are two types of magnetic poles: N-pole (north pole) and S-pole (south pole).
Magnetic monopoles do not exist, so magnetic force cannot be formulated in a way analogous to the electric or gravitational forces. But similar to the masses and electric charges which create vector field in space (gravitational and electric fields), a magnet creates a vector field in space that is called magnetic field. The intensity of the magnetic field at any given point in space is usually represented by vector 'B'.
In addition to the permanent magnets, magnetic field can be created by charge carrying wires, or as a consequence of any change in a electric field in space.

Magnetic Field Lines

Similar to the electric field, magnetic field can be represented by magnetic field lines. Since isolated magnetic poles (or magnetic monopoles) do not exist in the world, the magnetic field lines leave the N-pole of a magnet, and enter the S-pole, and they continue to form a closed loop inside the magnet. So the magnetic field lines are always associated with both poles of the magnet (or the magnetic dipole). Again similar to the electric field, direction of the magnetic field at a given point is tangent to the magnetic field line at that point, that is a direction in which the N-pole of a compass would point when placed at that point; and the magnetic field intensity at that point is proportional to the number of magnetic field lines passing through the unit area perpendicular to the field lines at that point. Magnetic field lines of some magnet arrangements, are shown in figures below:

Magnetic field lines are closed loops; they leave N-pole, and enter to the S-pole.

Magnetic field line pattern of identical opposite poles.

Magnetic Force Acting on a Charge in a Magnetic Field

If the electric charge 'q' is moving with velocity of 'V' in a magnetic field with intensity of 'B', the magnetic force (or electromagnetic force) acting on it, is given by this equation:

So the magnetic force acting on a moving electric charge is perpendicular to the both magnetic field intensity and the velocity of electric charge, and its direction depending on the sign of the electric charge, could be as follows:

So the magnetic force is a centripetal force with magnitude of,

where 'V' is the speed of charge, and 'B' is the magnitude of magnetic field.

Magnetic Force Acting on a Charge Carrying Wire

Assume a wire that carrying constant electric current 'I' is located in a magnetic field with intensity of 'B'.

A charge carrying wire in a magnetic field.

Magnetic force acting on the electric charge 'q' that is moving with velocity 'V' on wire, is given by,

since 'q' is positive, the magnitude of this force is,

where θ is the angle between 'V' (or direction of electric current) and 'B'.
If the length of the wire inside magnetic field is 'L', and the charge travels this distance in time 't', the speed of charge 'q' is V = L/t , and the equation above can be written as,

and since q/t=I, the equation above can be written as,

This force is maximum when magnetic field is perpendicular to the wire (or sin θ=1).
The SI unit of magnetic field intensity is Tesla 'T'. According to the equation above, if a wire carrying one Ampere electric current, is perpendicular to a magnetic field, where the magnetic force acting on one meter of the wire is one Newton, the magnetic field intensity is one Tesla.