#jsDisabledContent { display:none; } My Account | Register | Help

# Electromagnetic four-potential

Article Id: WHEBN0000731884
Reproduction Date:

 Title: Electromagnetic four-potential Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Electromagnetic four-potential

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector. [1]

As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant.

Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.

In this article, index notation and the Minkowski metric (+−−−) will be used, see also Ricci calculus, covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units.

## Contents

• Definition 1
• In the Lorenz gauge 2
• References 4

## Definition

The electromagnetic four-potential can be defined as:[2]

SI units cgs units
A^\alpha = \left( \phi / c , \mathbf{A} \right)\,\! A^\alpha = (\phi, \mathbf{A})

in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m−1 in SI, and Mx·cm−1 in Gaussian-cgs.

The electric and magnetic fields associated with these four-potentials are:[3]

SI units cgs units
\mathbf{E} = -\mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t} \mathbf{E} = -\mathbf{\nabla} \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t}
\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}. \mathbf{B} = \mathbf{\nabla} \times \mathbf{A}.

In special relativity, the electric and magnetic fields must be written in the form of a tensor so they transform correctly under Lorentz transformations - achieved by the electromagnetic tensor. This is written in terms of the electromagnetic four-potential as:

F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

## In the Lorenz gauge

Often, the Lorenz gauge condition \partial_{\alpha} A^{\alpha} = 0 in an inertial frame of reference is employed to simplify Maxwell's equations as:[4]

SI units cgs units
\Box A^\alpha = \mu_0 J^\alpha \Box A^\alpha = \frac{4 \pi}{c} J^\alpha

where Jα are the components of the four-current, and

\Box = \frac{1}{c^2} \frac{\partial^2} {\partial t^2}-\nabla^2

is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI units cgs units
\Box \phi =- \frac{\rho}{\epsilon_0} \Box \phi = 4 \pi \rho
\Box \mathbf{A} =- \mu_0 \mathbf{j} \Box \mathbf{A} = \frac{4 \pi}{c} \mathbf{j}

For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are:[5]

\phi (\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|}
\mathbf A (\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\mathbf{j}( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|},

where

t_r = t - \frac{\left|\mathbf{r}-\mathbf{r}'\right|}{c}

is the retarded time. This is sometimes also expressed with

\rho(\mathbf{r}',t_r)=[\rho(\mathbf{r}',t)],

where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying according to r −2 (the induction field) and a component decreasing as r −1 (the radiation field).

## References

1. ^ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
2. ^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
3. ^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
4. ^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
5. ^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
• Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press.
• Jackson, J D (1999). Classical Electrodynamics (3rd). New York: Wiley. ISBN ISBN 0-471-30932-X.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.