1. The problem statement, all variables and given/known data I'm trying to figure out the heat dissipation in a volume V due to an incident harmonic electric field. I know [tex] Q = \frac{1}{2}\int_V\mathrm{Re}\left(\mathbf{j}^{*} \cdot \mathbf{E}\right) d^3x [/tex] [tex] = \frac{1}{2}\int_V\mathrm{Re}\left(\left(\sigma \mathbf{E}\right)^{*} \cdot \mathbf{E}\right) d^3x [/tex] 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.
It depends, is [itex]\sigma[/itex] a scalar or a tensor? If it's a scalar, this is pretty straightforward. Since [itex]\sigma \mathbf{E}^* = ( \sigma E_x^*, \sigma E_y^*, \sigma E_z^* ) [/itex], try dotting that with E and see what you get.
Sigma (the complex conductivity) is a scalar. Anyway, I think this dot product should be [tex] \sigma(E_x^2,E_y^2,E_z^2) [/tex] but it seems weird to have E^2 as opposed to |E|^2 for complex numbers.
[tex] ( \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) [/tex] [tex] = ( \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) [/tex]
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?).
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.
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