# Fresnel Transmission Coefficient for Magnetic Field

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?

• Isnt your Helmholtz equation for free space with boundary conditions? So if are two media you need to write down a different equation. Jul 23, 2019 at 5:56

This is because the relationship between the E-field and H-field magnitudes is $$E = \eta H$$, where $$\eta$$ is the impedance.
Thus for two media with impedances $$\eta_1$$ and $$\eta_2$$, the "transmission coefficient for the H-field would be $$H_2 = T\frac{\eta_1}{\eta_2} H_1$$