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Electromagnetism is a theory developed by James Maxwell to explain the interrelationship between electricity and magnetism. At the heart of this theory is the notion of an electromagnetic field.

A stationary electromagnetic field stays bound to its origin. Examples of stationary fields are: the magnetic field around a wire or the electric field between the plates of a capacitor.

A changing electromagnetic field propagates away from its origin. The field forms a wave. Examples of dynamic fields: a ray of light, microwaves, and x-rays.

Different frequencies of electromagnetic waves describe color, radio waves, x-rays, etc. Collected under the name of Electromagnetic radiation. The behaviour of light is described by the field of optics.

Electromagnetism is a manifestation of the electromagnetic interaction (in particle physics.)

The whole of electromagnetism is governed by Maxwells equations, which are compatible with and served as a motivation for the theory of relativity.

Mathematical Description

The electromagnetic field exerts a force on charged particles. The formula for that force is:

F = qE + qv×B

where all boldfaced quantities are vectors: F is the force that a charge q experiences, E is the electric field at q's location, v is q's velocity, and B is the strength of the magnetic field at q's position.

This description of the force between charged particles, unlike Coulomb's force law, does not break down under relativity and in fact, the magnetic force is seen as part of the relativistic interaction of fast moving charges that Coulomb's law neglects.

The Electric Field E

The electric field E is defined such that, on a stationary charge:

F = qoE

where qo is what is known as a test charge. The size of the charge doesn't really matter, as long as it is small enough as to not influence the electric field by its mere presence. What is plain from this definition, though, is that the units of E are N/C, or Newtons per Coulomb.

The above definition seems a little bit circular, but in electrostatics, where charges are not moving, Coulomb's law works fine. So what we end up with is:

E = ∑qi(r - ri)(4πεo |r - ri|3)-1

where n is the number of charges, qi is the amount of charge associated with the 'i'th charge, ri is the position of the 'i'th charge, r is the position where the electric field is being determined, and εo is a universal constant called the permittivity of free space.

Note: the above is just coulomb's law, divided by q1, added up more multiple charges.

Changing the summation to an integral yields the following:

E = ∫ρrunit (4πεor2)-1dV

where ρ is the charge density as a function of position, runit is the unit vector pointing from dV to the point in space E is being calculated at, and r is the distance from the point E is being calculated at to the point charge.

Both of the above equations are extremely cumbersom, especially if one wants to calculate E as a function of position. There is, however, a scalar function called the electrical potential that can help. Electric potential, also called voltage (the units for which are the volt), which is defined thus:

φE = -∫E·ds

where φE is the electric potential, and s is the path over which the integral is being taken.

Unfortuneately, this definition has a caveat. In order for a potential to exist ∇×E must be zero. Whenever the charges are stationary, however, this condition will be met, and finding the field of a moving charge simply requires a relativistic transform of the electric field.

From the definition of charge, it is trivial to show that the electric potential of a point charge as a function of position is:

φ = q (4πεo|r - rq|)-1

where q is the point charge's charge, r is the position, and rq is the position of the point charge. The potential for a general distribution of charge ends up being:

φ = ∫ρr-1dV

where ρ is the charge density as a function of position, and r is the distance from the volume element dV.

Note well that φ is a scalar, which means that it will add to other potential fields as a scalar. This makes makes it relatively easy to break complex problems down in to simple parts and add their potentials. Getting the electric field from the potential is just a matter of taking the definition of φ backwards:

E = -∇φ

Calculating E from φ is so much easier than calculating E from the charge density that the electric field is more frequently expressed in V/m (volts per meter) than in Newtons per coulomb.