Electromagnetic Energy Density

[csrobertson]

Homework Statement

Find the electromagnetic energy and momentum density and their ratio. \\Determine the instantaneous values of the energy density, momentum density, intensity, and stress associated with the electromagnetic fields for the following cases. Also, identify these cases. Assume that ${\bf D} = \epsilon{\bf E}$ and ${\bf B} = \mu {\bf H}$.}

Case 1:
\begin{flalign}
&\begin{cases}
{\bf{E}} = E_oe_2e^{i(k_1x_1-\omega t)}\\
{\bf{H}} = \frac{k}{\mu \omega}\vec{k} \times {\bf{E}}
\end{cases}
\mbox{ for } k_1 = \sqrt{\epsilon\mu}\omega, k_2=k_2=0.&
\end{flalign}

Case 2: Same as case 1 with k_1 = i\alpha

Case 3:
\begin{flalign}
&\begin{cases}
B_r = \frac{\mu_om}{2\pi}\frac{\cos\theta}{r^3}\\
B_{\theta} = \frac{\mu_om}{4\pi}\frac{\sin\theta}{r^3}
\end{cases}&
\end{flalign}

Homework Equations

Electromagnetic Energy Density:

\mu = 1/2 (E\cdot D + B\cdot H)

The Attempt at a Solution

I have attempted making the substitutions into the above relevant equation and going through the math, however I am stuck on taking the dot product of exponential vectors. This might be something simple I am missing, but I have searched for hours online and have not been able to find a 'rule' for computing this value. Basically I need to know if:

E_oe_2e^{i(k_1x_1-\omega t)} \cdot E_oe_2e^{i(k_1x_1-\omega t)} = E_o^2 e_2^2 e^{2i(k_1x_1-\omega t}

*I apologize for LaTeX notation, for simplicity this is the general question I have:*

Does e^X * e^X = e^2X when X is a complex vector quantity, and * denotes the dot product.

 

[BigJDubs]

For reference, the other equations are as follows:Electromagnetic Momentum Density:\pi = {1\over c} (E \times H)Intensity:I = {1\over 2} c (E^2 + H^2)Stress:T= {1\over 2} (D \cdot E + B \cdot H)