D and E at boundary (dielectrics)

[mishima]

Homework Statement

Electromagnetics, Kraus, 4th edition problem 4.7.3
The y-z plane is the boundary between 2 dielectrics of relative permittivities ε_r = 2 and ε_r = 5. For negative values of x, E = (3,0,2) V/m. Find D (magnitude and direction) for positive values of x.

Homework Equations

The one that is similar to Snell's Law, not sure of name:
i) ε₂ tan (incident angle to normal) = ε₁ tan (refracted angle to normal)

which is derived from boundary conditions for 2 dielectrics with the absence of surface charges

ii)D1(normal) = D2(normal)
iii)E1(tangent) = E2(tangent)

The Attempt at a Solution

The x-axis is parallel to the normal of the boundary. So, I first found the angle the electric field vector makes with the x-axis to be 0.588 radians.

incident angle to normal (x-axis) = inverse tan (2/3)

This value was substituted into equation i) along with 5 and 2 for ε₁ and ε₂ respectively. Solving for the refracted angle, I obtained 1.03037 radians. I believe all of this is correct.

What I am having trouble with is now getting the magnitude of D2. If I just use equations ii) and iii) with pythagorean theorem I get an answer different than the book, which says: D2 = ε₀(6,0,10).

 

[mishima]

I figured it out, I had originally put the relative permittivity of 5 on the negative x side, but making it positive gives the correct result. Also, the "snell's law"-like equation isn't necessary at all.