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Electromagnetism
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Electricity · Magnetism
Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Electromagnetic four-potential

The electromagnetic four-potential is a covariant four-vector consisting of the electric potential and magnetic vector potential. All formulas in this article are given in SI units, and in parentheses in Gaussian-cgs units. The definition of the electromagnetic four-potential is:

A^{\alpha} = \left(\frac{\phi}{c}, \mathbf A \right) \qquad \left(A^{\alpha} = ( \phi, \mathbf A)\right)

in which φ is the electrical potential, and \mathbf A is the magnetic potential, a vector potential.

The units of Aα are volt·seconds/meter in SI, and maxwell/centimeter in Gaussian-cgs.

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

\mathbf{E} = -\mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t} \qquad \left( -\mathbf{\nabla} \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} \right)
\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}

It is useful to group the potentials together in this form because Aα is a contravariant vector, hence Aα = gαβAβ, A_\alpha=(\phi,-\mathbf{A}) is a covariant vector. This means that Aα transforms in the same way as the gradient of a scalar-valued function, e.g. \frac{\partial \psi}{\partial x^{\alpha}}\,, under arbitrary curvilinear coordinate transformations. So, for example, the inner product

g_{\alpha \beta} A^\alpha A^{\beta} = \frac{\phi^2}{c^2} - |\mathbf{A}|^2\qquad \left(g_{\alpha \beta} A^\alpha A^\beta \, = \phi^2 -|\mathbf{A}|^2 \right)

is the same in every inertial frame of reference.

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

\Box A^{\alpha} = \mu_0 J^{\alpha} \qquad \left( \Box A^{\alpha} = \frac{4 \pi}{c} J^{\alpha} \right)

where J^{\alpha} \, 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:

\Box \phi = \frac{\rho}{\epsilon_0} \qquad \left(\Box \phi = 4 \pi \rho \right)
\Box \mathbf{A} = \mu_0 \mathbf{j} \qquad \left( \Box \mathbf{A} = \frac{4 \pi}{c} \mathbf{j} \right)

For a given charge and current distribution, \rho(\mathbf{x},t) and \mathbf{j}(\mathbf{x},t), the solutions to these equations in SI units are

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

where \tau = t - \frac{\left|\mathbf{x}-\mathbf{x}'\right|}{c} is the retarded time. This is sometimes also expressed with \rho(\mathbf{x}',\tau)=[\rho(\mathbf{x}',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 as r − 2 (the induction field) and a component decreasing as r − 1 (the radiation field).

References

  • Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5.  
  • Jackson, J D (1999). Classical Electrodynamics (3rd). New York: Wiley. ISBN ISBN 0-471-30932-X.  

See also

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