Tensor and Electromagnetic Field

Maxwell’s Equations

are conclusions of experimental observations. The first equation shows the reality of never observing a single magnetic pole. The second one is a differential form of Faraday’s electromagnetic law. The third is Gauss’s law that derived from Coulomb’s law. The last one states the expansion of Ampere’s law (Reitz, 1979).

Electromagnetic field vectors E and B if written in potential scalar and vector will be

E and B aren’t four-vectors, despite their six components, which are: E1, E2, E3, B1, B2, B3, can be used to define anti symmetry tensors, with this way

or

And the same things are true for E2 and E3.

For B components:

We can use the same way for B2 and B3.

By defining a tensor

Then: iE1 = F41, B1 = F23, and so on.

Calculation of all tensor {F} components that have Fμν elements are

By this definition, this tensor

should be an antisymmetry tensor, because Fμν = - Fνμ and Fμμ = 0.

By using this electromagnetic field tensor, Maxwell’s equations can be expressed in a good relativistic notation

or

or

or

or

If we select 1, 2, 3 for λ, μ, ν, then this equation

will be

or

which is the Maxwell’s equation

Similarly, when the set of indices λ, μ are taken by 1, 2, 3 combination and ν = 4, the following component can be obtained

which is the equation

For example, when selected λ = 1, μ = 2 and ν = 4, then this equation

will be

or

Thus, two homogeneous Maxwell’s equations can be expressed only by this equation

And two inhomogeneous Maxwell’s equations can be obtained from

or

or

or

where

If we choose μ = 1, then this equation

will be

or

Also if we select μ = 2 and 3. Generally we can state

For μ = 4

where J4 = icρ, so

Thus, the four Maxwell equations can be expressed only by two equations involving the operation of field tensor components (Marion, 1995).