User Tools

Site Tools


maxwell_equations

Maxwell's equations

Stan Zurek, Maxwell's equations, Encyclopedia Magnetica,
http://www.e-magnetica.pl/doku.php/maxwell_equations

* This page is being edited and may be incomplete or incorrect.

Maxwell's equations or Maxwell-Heaviside equations - a set of mathematical equations which describe the behaviour of electromagnetic field in space and time.1)2)3)4)5)6)7)

In his original publication in 1865,8) James Clerk Maxwell listed 20 equations, which were split for each orthogonal coordinates (hence the large number of equations). These equations were later rationalised by Oliver Heaviside, who expressed them in a vector form which is know today.9) There are four basic electromagnetic equations, supplemented by additional constitutive relations, as well as expressions of energy conservation, and force.

To fully quantify an electromagnetic vector field it is necessary to determine its divergence as well as curl.10) For other fields, such as thermal, it is also useful to calculate gradient.

Electromagnetic quantities

The Maxwell-Heaviside equations make use of several basic electromagnetic quantities, which mathematically represent vector fields.

Electromagnetic quantities expressed as vector fields 11)12)
name symbol SI unit CGS unit
magnetic Magnetic flux density $$ \vec{B} $$ (T) ≡ (V·s / m2) (G)
Magnetic field strength $$ \vec{H} $$ (A/m) (Oe)
Magnetisation $$ \vec{M} $$ (A/m) (Oe)
Magnetic polarisation $$ \vec{P} $$ (T) (G)
electric Electric field $$ \vec{E} $$ (V/m) (statV/cm)
Electric displacement field $$ \vec{D} $$ (C/m2) (statV/cm)
Electric polarisation $$ \vec{P} $$ (C/m2) (statV/cm)

Maxwell's equations in general

From the viewpoint of theoretical physics, the equations can be expressed in a form which is always valid, in vacuum or in material. However, this requires a full knowledge of microscopic magnetic moments (at the level of subatomic particles) which makes their direct application very difficult for matter, which comprises of a very larger number of atoms.13)

Maxwell's equations, valid in general14)
differential form integral form
Gauss's law for electric field $$ ∇·\vec{E} = \frac{ρ}{ε_0} $$ $$ \oint_S \vec{E} · d\vec{a} = \frac{q}{ε_0} $$
Faraday's law of induction $$ ∇×\vec{E} = -\frac{∂\vec{B}}{∂t} $$ $$ \oint_C \vec{E}·d\vec{l} = - \frac{d}{dt}· \int_S \vec{B}·d\vec{a} $$
Gauss's law for magnetic field $$ ∇·\vec{B} = 0 $$ $$ \oint_S \vec{B}·d\vec{a} = 0 $$
Ampère-Maxwell's circuital law $$ ∇×\vec{B} = μ_0 · \vec{J} + μ_0 · ε_0 · \frac{∂\vec{E}}{∂t} $$ $$ \oint_C \vec{B}·d\vec{l} = μ_0 · I + μ_0 · ε_0 · \frac{d}{dt}· \int_S \vec{E}·d\vec{a} $$
where: ρ - electric charge density (C/m3), ε0 - electric permittivity of vacuum (F/m), q - electric charge (C), l - increment of path for integral (m), a - increment of surface for integral (m2), μ0 - magnetic permeability of vacuum (H/m), J - electric current density (A/m2), I - electric current (A), S - closed surface (region of integral), C - closed curve (path of integral)

Maxwell's equations in matter

In matter, there are localised magnetic moments which respond to the magnetic field penetrating the matter. It is possible to express the response of the matter as a vector field which is averaged (smoothed out) over the whole volume of the material, so that the vector field is expressed in effect as a macroscopic quantity, rather than microscopic variation (which can very wildly).15) These averaged out quantities are measurable experimentally on a macroscopic scale, hence useful for a direct experimental verification and technical purposes.

However, this approach requires further information about the relationship between the excitation and response of the matter, which can be quantified for example in the form of the magnetic permeability μ or electric permittivity ε.16)

Maxwell's equations, valid in matter17)
differential form integral form
Gauss's law for electric field $$ ∇·\vec{D} = ρ $$ $$ \oint_S \vec{D} · d\vec{a} = Q $$
Faraday's law of induction $$ ∇×\vec{E} = -\frac{∂\vec{B}}{∂t} $$ $$ \oint_C \vec{E}·d\vec{l} = - \frac{d}{dt}· \int_S \vec{B}·d\vec{a} $$
Gauss's law for magnetic field $$ ∇·\vec{B} = 0 $$ $$ \oint_S \vec{B}·d\vec{a} = 0 $$
Ampère-Maxwell's circuital law $$ ∇×\vec{H} = \vec{J} + \frac{∂\vec{D}}{∂t} $$ $$ \oint_C \vec{H}·d\vec{l} = I + \frac{d}{dt}· \int_S \vec{D}·d\vec{a} $$
where: ρ - electric charge density (C/m3), a - increment of surface for integral (m2), Q - electric charge (C), t - time (s), l - increment of path for integral (m), J - electric current density (A/m2), I - electric current (A), S - closed surface (region of integral), C - closed curve (path of integral)

Constitutive relationships

→ → →
Helpful page? Support us!
→ → →
PayPal
← ← ←
Help us with just $0.10 per month? Come on… ;-)
← ← ←

See also

References

This website uses cookies. By using the website, you agree with storing cookies on your computer. Also you acknowledge that you have read and understand our Privacy Policy. If you do not agree leave the website.More information about cookies
maxwell_equations.txt · Last modified: 2023/12/19 15:27 by stan_zurek

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki
Legal disclaimer: Information provided here is only for educational purposes. Accuracy is not guaranteed or implied. In no event the providers can be held liable to any party for direct, indirect, special, incidental, or consequential damages arising out of the use of this data.

For information on the cookies used on this site refer to Privacy policy and Cookies.