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## Homework Statement

The permittivity of a given medium is given by the equation

[tex]ε =

\begin{pmatrix}

ε_1 & 0 & 0\\

0 & ε_1 & 0\\

0 & 0 & ε_2

\end{pmatrix}

[/tex]

A wave is traveling in the [itex]\hat{k}[/itex] direction in this medium where the unit vector [itex]\hat{k}[/itex] is defined as [itex]\hat{k} = \hat{x}sin(θ)+\hat{z}cos(θ)[/itex]. All fields are proportional to [itex]e^{-i*(\overline{k}\bullet\overline{r})}[/itex] and [itex]\overline{E} = \overline{E_0}e^{-i*(\overline{k}\bullet\overline{r})}[/itex] where [itex]\overline{E_0}[/itex] is a constant vector.

Show that:

**a.**[itex]\overline{E_0}, \overline{D_0}[/itex] and [itex]\hat{k}[/itex] are coplanar.

**b.**[itex]\overline{H_0}[/itex] is perpendicular to this plane.

**c.**[itex]\overline{D_0}[/itex] is perpendicular to [itex]\hat{k}[/itex].

## Homework Equations

Determinant of 3x3 matrix

Maxwell–Faraday equation

Cross Product

## The Attempt at a Solution

I'm seeing this as more of a math problem than a physics problem.

**a.)**This is what I am thinking. If I can show that [itex]\overline{E_0}, \overline{D_0}[/itex] and [itex]\hat{k}[/itex] vectors are linearly dependent then this proves that they are coplanar. I can do this by setting up a 3x3 matrix where each column corresponds to one of these vectors and then take the determinant of said matrix. If that determinant is equal to zero then it proves that they are coplanar. In this case, I need to know all of the vectors.

[itex]\hat{k} = \hat{x}sin(θ)+\hat{z}cos(θ)[/itex]

[itex] \overline{E_0} = E_x\hat{x} + E_y\hat{y} + E_z\hat{z}[/itex]

[itex]\overline{D} = ε\overline{E} = (ε_1E_x\hat{x} + ε_1E_y\hat{y} + ε_2E_z\hat{z})e^{-i*(\overline{k}\bullet\overline{r})} = \overline{D_0}e^{-i*(\overline{k}\bullet\overline{r})}[/itex]

so

[itex]\overline{D_0} = ε_1E_x\hat{x} + ε_1E_y\hat{y} + ε_2E_z\hat{z}[/itex]

Now we set up the equation and take the determinant like so

[tex]det

\begin{pmatrix}

E_x & ε_1E_x & sinθ\\

E_y & ε_1E_y & 0\\

E_z & ε_2E_z & cosθ

\end{pmatrix}

[/tex]

[itex]= E_x(ε_1E_ycosθ)-ε_1E_x(E_ycosθ)+sinθ(E_yε_2E_z-E_zε_1E_y)[/itex]

[itex]=E_yE_zsinθ(ε_2-ε_1)[/itex]

This last line here should equal zero to show that the these three vectors are coplanar. This is also where I run into a problem. If the vectors are coplanar then at least one of the variables above(excluding epsilons) must be equal to zero. Now I could just assert that E_y or E_z is equal to zero, its not like they can't be but beside asserting it, I don't know how to show that one of these variables is zero. I'm stuck here. There is obviously some piece of information I am lacking.

**b.)**For this part, I was thinking of using the cross product to figure out if vectors are perpendicular. The magnitude of the cross product is as follows:

[itex]|a \times b| = |a||b|sinθ[/itex]

In the case that vectors are perpendicular to each other, θ = 90 deg so

[itex]|a \times b| = |a||b|[/itex]

If I take the cross product of [itex]\overline{H_0}[/itex] and [itex]\overline{D_0}[/itex] and compute its magnitude, and then compute the magnitude of [itex]\overline{H_0}[/itex] and [itex]\overline{D_0}[/itex] separately and multiply, they should be equal. In other words:

[itex]|\overline{H_0} \times \overline{D_0}| = |\overline{H_0}||\overline{D_0}|[/itex]

To get [itex]\overline{H}[/itex] and ultimately [itex]\overline{H_0}[/itex], I think I just need to use a form of the Maxwell–Faraday equation and crunch the variables.

**c.)**This is a repeat of part b but using vectors [itex]\overline{D_0}[/itex] and [itex]\hat{k}[/itex]

My biggest concern is in part A. I feel that there is a piece of info that I am missing and it is impeding me from completing the other parts.

Thanks for the help!