# Electromagnetic field

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The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction, one of the four fundamental forces of nature. The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz Force Law.

From a classical point of view, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner, whereas from a quantum mechanical point of view, the field can be viewed as being composed of photons.

## Structure of the electromagnetic fieldEdit

The electromagnetic field may be viewed in two distinct ways.

### Continuous structureEdit

Classically, electric and magnetic fields are thought of as being produced by smooth motions of charged objects. For example, oscillating charges produce electric and magnetic fields that may be viewed in a 'smooth', continuous, wavelike manner. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For instance, 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.

### Discrete structureEdit

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 its frequency $\nu$ through the equation:

$E= \, h \, \nu$

where $h$ is Planck's constant, named in honour of Max Planck, and $\nu$ is the frequency of the photon . 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 that 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 describing the interaction of electromagnetic radiation with charged matter.

## Dynamics of the electromagnetic fieldEdit

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 realized 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. These interactions are described by Maxwell's equations and the Lorentz force law.

## The electromagnetic field as a feedback loopEdit

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
• the fields interact with each other
• 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

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.

## Mathematical descriptionEdit

Main article: Mathematical descriptions of the electromagnetic field

There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional vector fields. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as $\mathbf{E}(x, y, z, t)$ (electric field) and $\mathbf{B}(x, y, z, t)$ (magnetic field).

If only the electric field ($\mathbf{E}$) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field ($\mathbf B$) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

With the advent of special relativity, physical laws became susceptible to the formalism of tensors. Maxwell's equations can be written in tensor form, generally viewed by physicists as a more elegant means of expressing physical laws.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed in a vacuum by Maxwell's equations. In the vector field formalism, these are:

$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ (Gauss' Law - electrostatics)
$\nabla \cdot \mathbf{B} = 0$ (Gauss' Law - magnetostatics)
$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$ (Faraday's Law)
$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ (Ampère-Maxwell Law)

where $\rho$ is the charge density, which can (and often does) depend on time and position, $\epsilon_0$ is the permittivity of free space, $\mu_0$ is the permeability of free space, and $\mathbf J$ is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permitivity of free space with the permeability and permitivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

The Lorentz force law governs the interaction of the electromagnetic field with charged matter.

## Properties of the fieldEdit

### Reciprocal behaviour of electric and magnetic fieldsEdit

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 generator.

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 and run an electric motor.

### Light as an electromagnetic disturbanceEdit

Maxwell's equations take the following, free space, form in an area that is very far away from any charges or currents - that is where $\rho$ and $\mathbf J$ are zero.

$\nabla \cdot \mathbf{E} = 0$
$\nabla \cdot \mathbf{B} = 0$
$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$
$\nabla \times \mathbf{B} = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$

In the above, the substitution $\mu_0 \epsilon_0 = \frac{1}{c^2}$ has been made, where $c$ is the speed of light. Taking the curl of the last two equations, the result is as follows.

$\nabla \times \nabla \times \mathbf{E} = \nabla \left ( \nabla \cdot \mathbf E \right ) - \nabla^2 \mathbf E = \nabla \times \left ( -\frac {\partial \mathbf{B}}{\partial t} \right )$
$\nabla \times \nabla \times \mathbf{B} = \nabla \left ( \nabla \cdot \mathbf B \right ) - \nabla^2 \mathbf B = \nabla \times \left ( \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \right )$

However, the first two equations mean $\nabla \left ( \nabla \cdot \mathbf E \right ) = \nabla \left ( \nabla \cdot \mathbf B \right ) = 0$. So plugging this in, and moving the curls within the time derivates and then plugging in for the resultant curls, the result is as follows.

$- \nabla^2 \mathbf E = -\frac{\partial}{\partial t} \left (\nabla \times \mathbf{B} \right ) = -\frac{\partial}{\partial t} \left ( \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \right ) = - \frac{1}{c^2} \frac{\partial^2 \mathbf E}{\partial t^2}$
$- \nabla^2 \mathbf B = \frac{1}{c^2} \frac{\partial}{\partial t} \left ( \nabla \times \mathbf{E} \right ) = \frac{1}{c^2} \frac{\partial}{\partial t} \left ( -\frac {\partial \mathbf{B}}{\partial t} \right ) = - \frac{1}{c^2} \frac{\partial^2 \mathbf B}{\partial t^2}$

Or:

$\nabla^2 \mathbf E = \frac{1}{c^2} \frac{\partial^2 \mathbf E}{\partial t^2}$
$\nabla^2 \mathbf B = \frac{1}{c^2} \frac{\partial^2 \mathbf B}{\partial t^2}$

Or even:

$\Box^2 \mathbf E = 0$
$\Box^2 \mathbf B = 0$

In this last form, the $\Box^2$ is the d'Alembertian, which is $\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}$, so the last two forms are the same thing written in two different ways. These can be identified as wave equations, that is, valid electric fields and magnetic fields have an oscillatory form, such as a sinusoid, which result in wave behaviors. Moreover, the first two of the free space Maxwell's equations imply that the waves are transverse waves. The last two of the free space Maxwell's equations imply that the wave of the electric field is in phase with and perpendicular to the magnetic field wave. Moreover, the $c^2$ term represents the speed of the wave. So these electromagnetic waves travel at the speed of light. James Clerk Maxwell, after whom Maxwell's equations are named, suggested when he made these calculations that as these waves travel at the same speed as light, that light would actually be such a wave. His suggestion proved correct, and light is indeed an electromagnetic wave.

## Relation to and comparison with other physical fieldsEdit

Main article: Fundamental forces

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

### Electromagnetic and gravitational fieldsEdit

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'.

The relative strengths and ranges of the four interactions and other information are tabulated below:

Theory Interaction mediator Relative Magnitude Behavior Range
Chromodynamics Strong interaction gluon 1038 1 infinite
Electrodynamics Electromagnetic interaction photon 1036 1/r2 infinite
Flavordynamics Weak interaction W and Z bosons 1025 1/r5 to 1/r7 10-18 m
Geometrodynamics Gravitation graviton 100 1/r2 infinite

## ApplicationsEdit

Properties of the electromagnetic field are exploited in many areas of industry. The use of electromagnetic radiation is seen in various disciplines. For example, X-rays are high frequency electromagnetic radiation and are used in radio astronomy, radiography in medicine and radiometry in telecommunications. Other medical applications include laser therapy, which is an example of photomedicine. Applications of lasers are found in military devices such as laser-guided bombs, as well as more down to earth devices such as barcode readers and CD players. Something as simple as a relay in any electrical device uses an electromagnetic field to engage or to disengage the two different states of output (ie, when electricity is not applied, the metal strip will connect output A and B, but if electricity is applied, an electromagnetic field will be created and the metal strip will connect output A and C).