The magnetic phase change of an electromagnetic wave during reflection

[Dom Tesilbirth]

Homework Statement

An electromagnetic wave is incident on a water-air interface. The phase of the perpendicular component of the electric field, $E_\bot$, of the reflected wave into the water is found to remain the same for all angles of incidence. By what angle does the phase of the magnetic field H change?

Relevant Equations

$$\overrightarrow{B}=\dfrac{1}{v}\widehat{k}\times \overrightarrow{E}$$

If the question had mentioned $\overrightarrow{E}$ instead of $E_\bot$, then we could have used $\overrightarrow{B}=\dfrac{1}{v}\widehat{k}\times \overrightarrow{E}$ to get the direction of the magnetic field. But the question had only mentioned $E_\bot$. To my understanding, knowing $E_\bot$ is not enough. Because, according to $\overrightarrow{B}=\dfrac{1}{v}\widehat{k}\times \overrightarrow{E}=\dfrac{1}{v}\widehat{k}\times \left( \overrightarrow{E}_{\parallel}+\overrightarrow{E}_{\bot}\right)$, the change in $\overrightarrow{B}$ also depends on the change in $\overrightarrow{E_\parallel}$.Is there any other way to solve this problem?

 

[tech99]

I immediately stumbled across the attached document which should be helpful for you.
https://scholar.harvard.edu/files/schwartz/files/lecture15-refraction.pdf