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**Boundary conditions & time domain electromagnetic waves: does classical model fit?**

Consider two propagating media: a lossy dielectric medium and a lossless dielectric medium. Thus, the interface that separates them has two tangential components of electric field, one for each medium. One of them, the component of the lossy medium, decays with time, say, ET1 = C * exp (-a * t) * sin (b * t). The other one, by being in a lossless medium, does not decay with time: let's say ET2 = D * sin (d * t), for example.

Classical electromagnetism states that the tangential electric field must be continuous at the interface of separation the two media, ie: C * exp (-a * t) * sin (b * t) = D * sin (d * t) for all t> 0. However, these two equations will never equal for all t due to the factor exp (-a * t), present in Et1 and absent in ET2.

Conclusion: there will never be transmission of electromagnetic energy between two media.

Obviously I am wrong. But where?

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