User:Mpatel/sandbox/Electromagnetic field

An electromagnetic field is a physical field produced by electrically charged objects. It affects the behaviour of charges in the field and is also affected by them. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction, one of the four fundamental forces of nature.

The electromagnetic field is studied extensively in theoretical physics and 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 viewpoint, the electromagnetic field is regarded as a smooth, continuous field, propagated in a wavelike manner, whereas from a quantum mechanical perspective, the field is quantised, being viewed as composed of photons.

James Clerk Maxwell discovered in the 19th century that disturbances in the electromagnetic field travel at the speed of light, thus suggesting that light is an electromagnetic wave. This led to deep insights into the nature of light, it's properties and applications thereof.

Generation

Fig. 1 - Electrostatic fields produced by stationary charges.

Fig. 2 - A magnetic field (B) produced by moving charges (i.e. a current).

It was known since antiquity that charged objects produced forces. In modern terminology, any stationary charge produces an electrostatic field (Fig. 1) whereas a moving charge generates a magnetic field as well; the magnetic field is called a magnetostatic field if the current does not change over time. (Fig. 2)

Physical structure

The electromagnetic field may be viewed in two distinct ways.

Continuous structure

Main article: Classical electromagnetism

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 when the radiation has low frequency, but problems are found at high frequencies (see ultraviolet catastrophe). This problem leads to another view.

Discrete structure

Main articles: Photon and Quantum electrodynamics

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

Main articles: Maxwell's equations and Lorentz force law

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 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' (curl) 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.

Mathematical description

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 written in SI units as:

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$ (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. Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity 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

There are some important properties that the electromagnetic field possesses. These properties have many useful consequences.

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 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 disturbance

Fig. 3 - Depiction of light as an electromagnetic wave

One of the most fascinating discoveries in the history of science was the realisation that light is a form of electromagnetic radiation. This idea first gained validation when James Clerk Maxwell proved the result in his 1864 paper A Dynamical Theory of the Electromagnetic Field. Maxwell commented:

“ The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws. ”

Derivations

There are various ways of showing that disturbances in the electromagnetic field (in vacuum) propagate at the speed of light. Some are presented here:

Standard Vector Calculus Derivation

Assumptions:

• Maxwell's equations in vacuum with the electric and magnetic fields represented as vector fields.

In vacuum, Maxwell's equations take the following form in a region that is very far away from any charges or currents - that is where $\rho$ and $\mathbf {J}$ can be taken to be 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 results in:

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

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

However, the first two Maxwell equations imply $\nabla \left(\nabla \cdot \mathbf {E} \right)=\nabla \left(\nabla \cdot \mathbf {B} \right)=0$. Using this, and moving the curls within the time derivatives and then plugging in for the resultant curls, gives:

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

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

These equations become,

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

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

i.e.,

$\Box ^{2}\mathbf {E} =0$

$\Box ^{2}\mathbf {B} =0$

where $\Box ^{2}=\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}$ is the d'Alembertian. So, the last two forms are the same. These can be identified as wave equations, that is, valid electric 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.

Special Relativity Derivation

Assumptions:

• Postulates of special relativity
• Continuity equation.
• Maxwell's equations with the electromagnetic field represented as a 2-form field.

The four-vector formalism used in special relativity allows a remarkably simple derivation. The quantities $E$ and $B$ are the entries in a 4 by 4 skew-symmetric matrix called the electromagnetic field tensor $F^{ab}$. Starting with Maxwell's equations as formulated in special relativity

$F^{ab}{}_{,a}=\,\mu _{0}J^{b}$

and noting that $J^{b}=0$ expresses the vacuum condition, a partial derivative with respect to $x^{b}$ of the former equation yields,

$F^{ab}{}_{,ab}=\,0$

This last equation is a shorthand way of saying that the d'Alembertian of each entry of the electromagnetic field tensor is zero. Note that the vacuum condition is not required in this derivation.

Comparison with other physical fields

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 fields

Main article: Gravitomagnetism

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'. In addition, quantum mechanics stipulates that photon has a spin of 1, whereas the (as yet undiscovered) graviton has a spin of 2.

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

Interaction	Current Theory	Mediators	Relative Strength	Long-Distance Behavior	Range(m)
Strong	Quantum chromodynamics (QCD)	gluons	1038	${1}$	10-15
Electromagnetic	Quantum electrodynamics (QED)	photons	1036	${\frac {1}{r^{2}}}$	infinite
Weak	Electroweak Theory	W and Z bosons	1025	${\frac {e^{-m_{W,Z}r}}{r}}$	10-18
Gravitation	General Relativity (GR)	gravitons (Hypothetical)	1	${\frac {1}{r^{2}}}$	infinite

Applications

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).

Health and safety

The potential health effects of the very low frequency EMFs surrounding power lines and electrical devices are the subject of on-going research and a significant amount of public debate. In workplace environments, where EMF exposures can be up to 10,000 times greater than the average, the National Institute for Occupational Safety and Health (NIOSH) has issued some cautionary advisories but stresses that the data is currently too limited to draw good conclusions.

The potential effects of electromagnetic fields on human health vary widely depending on the frequency and intensity of the fields. For more information on the health effects due to specific parts of the electromagnetic spectrum, see the following articles: -

• Static electric fields: see Electric shock
• Static magnetic fields: see MRI/Safety for one of the few applications in which magnetic fields are strong enough to have safety implications
• Extremely low frequency (ELF): see Power lines/health concerns
• Radio frequency (RF): see Electromagnetic radiation and health
• Light: see Laser safety
• Ultraviolet (UV): see Sunburn
• Gamma rays: see Gamma ray
• Mobile telephony: see Mobile phone radiation and health

See also

• Afterglow plasma
• Antenna factor
• Classification of electromagnetic fields
• Electric field
• Electromagnetism
• Electromagnetic tensor
• Fundamental interaction
• Electromagnetic radiation
• Electromagnetic spectrum
• Gravitational field
• List of environment topics
• Magnetic field
• Maxwell's equations
• Photoelectric effect
• Photon
• Quantum electrodynamics

External links

• On the Electrodynamics of Moving Bodies by Albert Einstein, June 30, 1905.
  • On the Electrodynamics of Moving Bodies (pdf)
• Non-Ionizing Radiation, Part 1: Static and Extremely Low-Frequency (ELF) Electric and Magnetic Fields (2002) by the IARC.

• Report on the efficacy of electromagnetic screening for sports injuries
• National Institute for Occupational Safety and Health - EMF Topic Page
• Electric Magnetic Fields In Your Environment