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TE and TM Waves

Note well that we have written the mutilated Maxwell Equations so that the $ z$ components are all on the right hand side. If they are known functions, and if the only $ z$ dependence is the complex exponential (so we can do all the $ z$ -derivatives and just bring down a $ \pm ik$ ) then the transverse components $ \Vec{E}_\perp$ and $ \Vec{B}_\perp$ are determined!

In fact (for propagation in the $ +z$ direction, $ e^{+ikz - i\omega t}$ ): ik\Vec{E} _&perp#perp;+ i&omega#omega;(\Hat{z} ×\Vec{B} _&perp#perp;) & = & _&perp#perp;E_z
ik(\Hat{z} ×\Vec{E} _&perp#perp;) + i&omega#omega;\Hat{z} ×(\Hat{z} ×\Vec{B} _&perp#perp;) & = & \Hat{z} ×_&perp#perp;E_z
ik(\Hat{z} ×\Vec{E} _&perp#perp;) & = & i&omega#omega;\Vec{B} _&perp#perp;+ \Hat{z} ×_&perp#perp;E_z
\Hat{z} ·( \Vec{\nabla} _&perp#perp;×\Vec{B} _&perp#perp;) & = & -i&omega#omega;&mu#mu;&epsi#epsilon;E_z and ik\Vec{B} _&perp#perp;- i&omega#omega;&mu#mu;&epsi#epsilon;(\Hat{z} ×\Vec{E} _&perp#perp;) & = & _&perp#perp; B_z
ik\Vec{B} _&perp#perp;- _&perp#perp;B_z & = & i&omega#omega;&mu#mu;&epsi#epsilon; (\Hat{z} ×\Vec{E} _&perp#perp;)
ik^2&omega#omega;&mu#mu;&epsi#epsilon;\Vec{B} _&perp#perp;- k&omega#omega;&mu#mu;&epsi#epsilon;_&perp#perp;B_z & = & ik (\Hat{z} ×\Vec{E} _&perp#perp;)
ik^2&omega#omega;&mu#mu;&epsi#epsilon;\Vec{B} _&perp#perp;- k&omega#omega;&mu#mu;&epsi#epsilon;_&perp#perp;B_z & = & i&omega#omega;\Vec{B} _&perp#perp;+ \Hat{z} ×_&perp#perp;E_z
or \Vec{B} _&perp#perp;& = & i&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2( k_&perp#perp; B_z + &mu#mu;&epsi#epsilon;&omega#omega;(\Hat{z} ×_&perp#perp;E_z))
\Vec{E} _&perp#perp;& = & i&mu#mu;&epsi#epsilon;&omega#omega;^2 - k^2( k_&perp#perp; E_z - &omega#omega;(\Hat{z} ×_&perp#perp;B_z)) (where we started with the second equation and eliminated $ \Hat{z}\times\Vec{B}_\perp$ to get the second equation just like the first).

Now comes the relatively tricky part. Recall the boundary conditions for a perfect conductor: \Hat{n} ×(\Vec{E} - \Vec{E} _c) = \Hat{n} ×\Vec{E} & = & 0
\Hat{n} ·(\Vec{B} - \Vec{B} _c) = \Hat{n} ·\Vec{B} & = & 0
\Hat{n} ×\Vec{H} & = & \Vec{K}
\Hat{n} ·\Vec{D} & = & &Sigma#Sigma; They tell us basically that $ \Vec{E}$ ($ \Vec{D}$ ) is strictly perpendicular to the surface and that $ \Vec{B}$ ($ \Vec{H}$ ) is strictly parallel to the surface of the conductor at the surface of the conductor.

This means that it is not necessary for $ E_z$ or $ B_z$ both to vanish everywhere inside the dielectric (although both can, of course, and result in a TEM wave or no wave at all). All that is strictly required by the boundary conditions is for E_z|_S = 0 on the conducting surface $ S$ (it can only have a normal component so the $ z$ component must vanish). The condition on $ B_z$ is even weaker. It must lie parallel to the surface and be continuous across the surface (where $ \Vec{H}$ can discontinuously change because of $ \Vec{K}$ ). That is: B_zn|_S = 0

We therefore have two possibilities for non-zero $ E_z$ or $ B_z$ that can act as source term in the mutilated Maxwell Equations.



Subsections
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Next: TM Waves Up: Wave Guides Previous: TEM Waves   Contents
Robert G. Brown 2017-07-11