TEM mode solution of Maxwell's equations
Maxwell's equations in a space among conductors of transmission line are:
The transverse electric and magnetic (TEM) mode solutions are:
where the suffix T indicates the transverse components:
The scalar potential ƒÓ satisfies the 2D Laplace equation and the boundary condition:

We will derive the TEM mode solution in a transmission line.
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In a space among conductors of transmission line, electric charges or currents do not exist. Then Maxwell's equations are:
Decomposing the fields into frequency components by Fourier transformation:
we obtain Maxwell's equations in the frequency domain:
Note that the last two equations automatically hold because of the first two equations.
For convenience, decomposing the fields into the transverse (suffix T) and longitudinal (suffix z) components:
, we obtain the four equations:
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Assuming the transverse electric and magnetic (TEM) mode :
, the equations are simplified:
Eliminating B_{T} between (2) and (4):
, we finally obtain the wave equation for E_{T}. Noting eq (1), E_{T} can be expressed as a gradient of a scalar potential ƒÓ(x,y):
Substituting this into the wave equation:
, we obtain the TEM mode solution for E_{T}.
Noting that the divergence of E is zero, the scalar potential ƒÓ(x,y) must satisfy the 2DLaplace's equation:
Note that u is the propagation speed:
Substituting the solution E_{T} into (2), we obtain B_{T }:
The directions of the electromagnetic fields_{ }and propagation direction z are depicted as follows: