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Electromagnetism

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Electromagnetism

Electromagnetism is a theory unified by James Clerk 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 carrying current or the electric field between the plates of a capacitor.

A changing electromagnetic field propagates away from its origin in the form of a wave. These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves, microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle physics this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.

The subfield of electromagnetism dealing specifically with the rapidly changing electric and magnetic fields which constitute light, is called electrodynamics.

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

Mathematical Description

The electromagnetic field exerts the following force (often called the Lorentz force) on charged particles:

$\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}$

$\mathbf{F} = q \mathbf{E} + q \frac{\mathbf{v}} {c} \times \mathbf{B}$

in Gauss units,

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, B is the strength of the magnetic field at q's position, and c is the speed of light.

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:

$\mathbf{F} = q_o \mathbf{E}$

where q_o 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 unit of E is N/C, or newtons per coulomb. This unit is equal to V/m (volts per meter), see below.

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:

where n is the number of charges, q_i is the amount of charge associated with the 'i'th charge, r_i is the position of the 'i'th charge, r is the position where the electric field is being determined, and ε_o 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:

$E = \int \rho \mathbf{r}_{unit} (4 \pi \epsilon_o r^2) ^{-1} dV$

where ρ is the charge density as a function of position, r_unit 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 cumbersome, 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:

$\phi_\mathbf{E} = - \int_s \mathbf{E} \cdot d\mathbf{s}$

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

Unfortunately, this definition has a caveat. In order for a potential to exist $\nabla \times \mathbf{E}$ 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:

$\phi = q (4 \pi \epsilon_o \left| \mathbf{r} - \mathbf{r}_q \right|)^{-1}$

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

$\phi = (4 \pi \epsilon_o)^{-1} \int \rho r^{-1} dV$

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 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:

$\mathbf{E} = -\nabla \phi$

From this formula it is clear that E can be expressed in V/m (volts per meter).

Electromagnetic Method

A geophysical method in which the magnetic and or electric fields resulting from generated surface currents are measured. Measurements may be made in the frequency domain at a number of frequencies, or the time domain at several time intervals after a transient pulse. Natural field methods such as magnetotellurics (MT) use natural magnetic and electromagnetic field as the source.

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