Exercise 3.4.1: Refraction of E at a dielectric interface

Exercise 3.4.1: Refraction of \vec E at a dielectric interface

The ten exercises below are in the spirit of Problem 3.4.1, the home exercises and the worked examples. They use only $D = ε_{0} E + P = ε E$ , Gauss's law $\oint_{S} D \cdot d S = Q_{free}$ , the boundary conditions, the bound charges, $C = Q / V$ , $W = \frac{1}{2} \int D \cdot E d v$ and the image principle.

Exercise 3.4.1

Refraction of $E$ at a dielectric interface

cf. boundary conditions

The plane $z = 0$ separates two simple dielectrics: $ε_{r 1} = 2$ for $z < 0$ and $ε_{r 2} = 5$ for $z > 0$ . There is no free charge on the interface. In region 1, just below the interface,

E_{1} = 2 a_{x} - 3 a_{y} + 5 a_{z} [V/m] .

Find (a) $E_{2}$ and $D_{2}$ just above it, (b) the angles $θ_{1}$ , $θ_{2}$ that $E_{1}$ , $E_{2}$ make with the normal, and (c) the bound surface charge density on the interface.

Interface z=0 between dielectrics 2 (below) and 5 (above), with E1 and E2 and the angles to the normal

Try it yourself first.