User:Mpatel/sandbox/Electromagnetic field

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The electromagnetic field is a physical influence (more precisely, a field) that is imagined to permeate through all of space and which arises from charged objects and describes one of the four fundamental forces of nature - electromagnetism. It can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by non-moving charges, the magnetic field by moving charges (currents), these two often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is dictated by Maxwell's equations and the Lorentz Force Law.

1 Nature of the electromagnetic field
2 Mathematical description
- 2.1 Vector field approach
- 2.2 Tensor field approach
3 Properties of the field
- 3.1 Reciprocal behaviour of electric and magnetic fields
- 3.2 Light as an electromagnetic disturbance
4 Relation to and comparison with other physical fields
5 Other descriptions of the electromagnetic field
- 5.1 Classical fluid interpretation
- 5.2 Photonic fluid interpretation
6 Everyday applications
7 The electromagnetic field as a feedback loop
8 See also
9 External links

[edit] Nature of the electromagnetic field

As with many physical concepts, there are various ways of thinking about the electromagnetic field. The field may be viewed in two distinct ways.

[edit] Continuous structure

Oscillating electric and magnetic fields may be viewed in a 'smooth', continuous, wavelike manner. These fields are sometimes assumed to vary sinusoidally with a single frequency. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For example, the metal atoms in a radio transmitter appear to transfer energy continuously. This view is useful to a certain extent (radiation of low frequency), but problems are found at high frequencies (see ultraviolet catastrophe). This problem leads to another view.

[edit] Discrete structure

The electromagnetic field may be thought of in a more 'coarse' way. Experiments reveal that electromagnetic energy transfer is better described as being carried away in 'packets' or 'chunks' called photons with a fixed frequency. Planck's relation links the energy $E$ of a photon to it's frequency $f$ through the equation:

$E= \, h f$

where $h$ is Planck's constant, named in honour of Max Planck. For example, in the photoelectric effect - the emission of electrons from metallic surfaces by electromagnetic radiation - it is found that increasing the intensity of the incident radiation has no effect and only the frequency of the radiation is relevant in ejecting electrons.

This quantum picture of the electromagnetic field has proved very successful, giving rise to quantum electrodynamics, a quantum field theory which describes the interaction of electromagnetic radiation with charged matter.

[edit] Dynamics

In the past, electrically charged objects were thought to produce two types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge and a magnetic field (as well as an electric field) is produced when the charge moves (creating an electric current) with respect to this observer. Over time, it was realised that the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field.

Once this electromagnetic field has been produced from a given charge distribution, other charged objects in this field will experience a force (in a similar way that planets experience a force in the gravitational field of the Sun). If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced. Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move and which is also affected by them.

Maxwells equations and the Lorentz Force Law describe how the electromagnetic field interacts with charged objects.

[edit] Mathematical description

There are different mathematical ways of representing the electromagnetic field.

[edit] Vector field approach

The electric and magnetic fields are usually described by the use of vector fields on $\R^3$ (3-space). These vector fields each have a value defined at every point of space and time and are thus to be regarded as functions of the space and time coordinates. As such, they are often written as $\vec{E}(x, y, z, t)$ (electric field) and $\vec{B}(x, y, z, t)$ (magnetic field).

If only the electric field ( $\vec{E}$ ) is non-zero, and is constant in time, the field is said to be an electrostatic field. The behaviour of electric and magnetic fields is governed by Maxwell's equations:

$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$ (Gauss' Law - electrostatics)

$\nabla \cdot \vec{B} = 0$ (Gauss' Law - magnetostatics)

$\nabla \times \vec{E} = -\frac {\partial \vec{B}}{\partial t}$ (Faraday's Law)

$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0\varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ (Ampère-Maxwell Law)

where <include notation>.

The electric and magnetic fields transform under a Lorentz boost in the direction $\vec{v}$ as:

$\vec{E}' = \gamma \left( \vec{E} + \vec{v} \times \vec{B} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \vec{E} \cdot \vec{v} ) \vec{v}$

$\vec{B}' = \gamma \left( \vec{B} - \frac {\vec{v} \times \vec{E}}{c^2} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \vec{B} \cdot \vec{v} ) \vec{v}$

[edit] Tensor field approach

The electric and magnetic fields can be combined together mathematically to form a bivector (4 $\times$ 4 anti-symmetric matrix) $\, F^{ab}$ , called the electromagnetic field tensor. This matrix effectively treats the electric and magnetic fields on the same footing and as a combined whole. Maxwell's equations may be written succinctly in terms of this bivector as:

$F_{[ab,c]} \, = 0$ and $F^{ab}{}_{,b} \, = \mu_0 J^a$

where <include notation>.

This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.

[edit] Properties of the field

[edit] Reciprocal behaviour of electric and magnetic fields

The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field. Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric motor.

The Ampère-Maxwell Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field.

[edit] Light as an electromagnetic disturbance

Being a dynamical field, it makes sense to inquire about the 'movement' of the field. More precisely, disturbances in the electromagnetic field transfer energy; how quickly this energy transfer takes place can be determined by use of Maxwell's equations. It is found that disturbances in the electromagnetic field propagate at the speed of light. Historically, Maxwell derived this result and made the suggestion that light is an electromagnetic wave. The modern stance is to view light as composed of photons, as this gives a more consistent description of the properties of light (for example, wave-particle duality and the photoelectric effect).

[edit] Relation to and comparison with other physical fields

Being one of the four fundamental forces of nature, it is useful to compare the electromagnetic field with the gravitational, strong and weak nuclear forces. The word 'force' is sometimes replaced by 'interaction'.

Sources of electromagnetic fields consist of two types of charge - positive and negative. This contrasts with the sources of the gravitational field, which are masses. Masses are sometimes described as 'gravitational charges', the important feature of them being that there is only one type (no 'negative masses'), or, in more colloquial terms, 'gravity is always attractive'.

[edit] Other descriptions of the electromagnetic field

[edit] Classical fluid interpretation

Main article: Hydrodynamic interpretation of the electromagnetic field

The electromagnetic field may be visualised in analogy to a fluid. In particular, the electric and magnetic vector fields can be thought of as being the velocities of a pair of incompressible fluids which permeate space. In the absence of charges, these fluids would be at rest, so that their velocity fields would be zero. Since both fluids are incompressible, their densities do not change: it is not possible to compress magnetic or electric fluid into a smaller space.

[edit] Photonic fluid interpretation

An alternative interpretation would be that the field is not actually a velocity field, but a flux density field of photonic fluid, which is constantly moving at the same speed: the speed of light, independent of the speed of the observer (the charged object). Photonic fluid never changes speed but can change net direction and the intensity of its net movement in that direction.

The velocity field interpretation is related to the hypothesis of a luminiferous aether through which electromagnetic waves would propagate. The proposition that the motion of the earth relative to the aether might be detectable (i.e. through an "aether wind") was disproven by the Michelson-Morley experiment, whereupon it was argued that the experiment had disproved the very existence of the aether. This opinion prevailed, but remains disputed by some who equate the classical concept of the aether with the modern notion of a quantum electrodynamic fluid. (The disputants argue that proving that the earth does not travel through an "aether wind" is no more nor less significant than proving that the earth does not travel through its own gravitational or magnetic fields.) The necessity of an aether was seen to have vanished when it was replaced by Einstein's theory of relativity.

According to special relativity, the Lorentz force equation reduces to the equation

$\mathbf{F} = q \mathbf{E}.$

The magnetic field becomes a relativistic by-product of the electric field, i.e. Lorentz transformations cause magnetic fields to be induced from electric fields, and vice versa. So the photonic fluid describes the electric field, and relativistic effects account for the derivative magnetic field. (This can be derived by applying a Lorentz transformation to a simplified version of Maxwell's equations, and it is mentioned by Einstein in his paper On The Electrodynamics Of Moving Bodies [1].)

The speed of light is invariant under a Lorentz transformation, but the velocity of light is changed. The component of the velocity of light parallel to the boost is left unchanged, but the transverse component is rotated: it is accelerated in a direction parallel to the boost. The addition of special relativity allows the combination of the electric and magnetic fields into a single tensor field. The tensor character of this combined electromagnetic field implies that the field is anisotropic with respect to the velocity of the charged particle on which it produces a force: the Lorentz force varies with the velocity of the charged particle.

[edit] Everyday applications

This section is a stub. You can help by expanding it.

Electromagnetic fields can be explained with a quantum basis by quantum electrodynamics.

According to special relativity, the Lorentz force equation reduces to the equation

$\mathbf{F} = q \mathbf{E}.$

[edit] The electromagnetic field as a feedback loop

The behavior of the electromagnetic field can be resolved into four different parts of a loop: (1) the electric and magnetic fields are generated by electric charges, (2) the electric and magnetic fields interact only with each other, (3) the electric and magnetic fields produce forces on electric charges, (4) the electric charges move in space.

The feedback loop can be summarized in a list, including phenomena belonging to each part of the loop:

charges generate fields
- Gauss's law Coulomb's law: charges generate electric fields
- Ampère's law: currents generate magnetic fields ( $\star$ )
the fields interact with each other
- displacement current: changing electric field acts like a current, generating vortex of magnetic field
- Faraday induction: changing magnetic field induces (negative) vortex of electric field
- Lenz's law: negative feedback loop between electric and magnetic fields
- Maxwell-Hertz equations: simplified version of Maxwell's equations
- electromagnetic wave equation
fields act upon charges
- Lorentz force: force due to electromagnetic field
  - electric force: same direction as electric field
  - magnetic force: perpendicular both to magnetic field and to velocity of charge ( $\star$ )
charges move
- continuity equation: current is movement of charges

Phenomena in the list are marked with a star ( $\star$ ) if they consist of magnetic fields and moving charges which can be reduced by suitable Lorentz transformations to electric fields and static charges. This means that the magnetic field ends up being (conceptually) reduced to an appendage of the electric field, i.e. something which interacts with reality only indirectly through the electric field.