How to Evaluate the Dot Product of Complex Electric Field Vectors?

[capandbells]

Homework Statement

I'm trying to figure out the heat dissipation in a volume V due to an incident harmonic electric field.
I know
<br /> Q = \frac{1}{2}\int_V\mathrm{Re}\left(\mathbf{j}^{*} \cdot \mathbf{E}\right) d^3x<br />
<br /> = \frac{1}{2}\int_V\mathrm{Re}\left(\left(\sigma \mathbf{E}\right)^{*} \cdot \mathbf{E}\right) d^3x<br />

My biggest problem is that I don't know how to evaluate that dot product. If someone could please either explain it or link me to a resource that explains working with complex vectors, I would appreciate it. I haven't taken an EM class yet and I've never had it explained elsewhere how to operate with complex vectors, so I am mostly flailing around with this stuff.

 

[Redbelly98]

It depends, is \sigma a scalar or a tensor?

If it's a scalar, this is pretty straightforward. Since \sigma \mathbf{E}^* = ( \sigma E_x^*, \sigma E_y^*, \sigma E_z^* ), try dotting that with E and see what you get.

 

[capandbells]

Sigma (the complex conductivity) is a scalar. Anyway, I think this dot product should be
<br /> \sigma(E_x^2,E_y^2,E_z^2)<br />
but it seems weird to have E^2 as opposed to |E|^2 for complex numbers.

 

[Redbelly98]

Sigma (the complex conductivity) is a scalar.

Okay, good.

Anyway, I think this dot product should be
<br /> \sigma(E_x^2,E_y^2,E_z^2)<br />

That's incorrect. Can you show how you came up with that?

 

[capandbells]

<br /> ( \sigma \mathbf{E})^{*} \cdot \mathbf{E} = ( ( \sigma E_x)^{*}, ( \sigma E_y)^{*}, ( \sigma E_x)^{*}) \cdot (E_x, E_y, E_z) = (( \sigma E_x)^{*})^{*}(Ex) + (( \sigma E_y)^{*})^{*}(E_y) +(( \sigma E_y)^{*})^{*}(E_z) <br />
<br /> = ( \sigma E_x)(E_x) + ( \sigma E_y)(E_y) + ( \sigma E_z)(E_z) = \sigma (E_x^2 + E_y^2 + E_z^2)<br />

 

[Redbelly98]

Okay.

The way I learned it, a dot product does not involve taking the complex conjugate of the first vector. So actually you would get terms like E_x*·E_x, etc., and you'd end up with σ|E|².

Note that heat dissipation Q should be a real number (right?).

 

[Antiphon]

The expression for energy invloves a conjugate so that energy is real. The dot product is a vector operation and doesn't care if the components are complex or not.

 

[Reloaded47]

lets just go from basics, step by step, assuming everything is complex

(sigma E)* . E
where E(vector) and sigma(scalar) are complex

(sigma)* E* . E = sigma* |E|^2

or

sigma* (Ex*, Ey*, Ez*).(Ex, Ey, Ez) = sigma* [|Ex|^2+|Ey|^2+|Ez|^2] = sigma* |E|^2

must take modulus of the field and its components