Helmholtz equations for electric and magnetic fields are
$$∇^2 \mathbf{H} + k^2 \mathbf{H} = \mathbf{0}$$ $$∇^2 \mathbf{E} + k^2 \mathbf{E} = \mathbf{0}$$
Obviously, if a solution is found to satisfy the electric field equation, it must also satisfy the magnetic field equation. A wave traveling between two media has an electric field magnitude in medium one proportion the magnitude in medium two, in other words
$$ |\mathbf E_2| = T |\mathbf E_1| $$
where $T$ is the Fresnel transmission coefficient. Is this true for the magnetic field as well?
$$ |\mathbf H_2| = T |\mathbf H_1| $$
If not why? How do we explain that the Helmholtz solution for electric and magnetic field could be the same?
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