# Plane Wave at Barrier

## Homework Statement

Sorry for the dull question. Problem is as shown/attached

## Homework Equations

The waves in part ii) are travelling in a HIL dielectric of permittivity ##\epsilon_{r}## from ##0 <z<d## and then hit an ideal metal boundary at ##z=d##.

## The Attempt at a Solution

I figure this should be quite obvious! The total field in the dielectric ##\vec{E_{D}}## is the superposition of the reflected and incident waves. At ##z=d## the field parallel to the boundary must be 0 -ideal metal is ideal conductor and so electrons can move to precisely counteract the field. We also know (from applying Faraday's law to a small loop around the boundary) that the parallel electric field must be continuous across the boundary. i.e

$$\vec{E_{d}} =E_{D}\hat{\mathbf{x}}= \bigg(E_{xi}exp[i(kz-\omega t)] + E_{xr} exp[i(-kz-\omega t) ]\bigg)\hat{\mathbf{x}}$$
[n.b - the frequency of the incident and reflected waves has to be the same, else we couldn't satisfy the boundary condition for all t,] So:
$$E_{D}(z=d) =0 \implies E_{xi} \big[cos(kd) +isin(kd)\big] =-E_{xr} \big[cos(kd)-isin(kd)\big]$$
$$E_{xr} = -\frac{ cos(kd)+isin(kd)}{cos(kd)-isin(kd)} E_{xi}$$
Which is obviously not the result I'm meant to get! What has gone wrong?