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Complex electric field vectors 
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#1
Jul211, 04:57 PM

P: 96

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. 


#2
Jul311, 02:54 PM

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P: 12,074

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. 


#3
Jul311, 04:10 PM

P: 96

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. 


#4
Jul311, 06:48 PM

Mentor
P: 12,074

Complex electric field vectors



#5
Jul311, 07:02 PM

P: 96

[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] 


#6
Jul311, 07:12 PM

Mentor
P: 12,074

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 σE^{2}. Note that heat dissipation Q should be a real number (right?). 


#7
Jul311, 08:48 PM

P: 1,781

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.



#8
Jul2011, 11:14 AM

P: 3

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 


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