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).