Negative Energy Density in EM Waves?

[jeffbarrington]

Homework Statement

The problem I have is that we are asked to show the complex relative permittivity of a good conductor is e_rc = 1 + i(sigma)/(omega*epsilon_0) where sigma is the conductivity and omega is the frequency of an electromagnetic wave in the medium. This is fine, I calculated it, the equation is given, it must be right. Now, we are also told that sigma is much greater than omega*epsilon_0, so the approximation it invites us to make is that e_rc = i(sigma)/(omega*epsilon_0). I think this is probably right so far.

The problem comes now. I am asked to find the energy density of the E-field in the wave. This is given by u = (1/2)E.D. Of course, this uses real values. I think D_c = e_rc*epsilon_0*E_c, and then I just take the real part of D_c to get D. However, the complex relative permittivity (assumed totally imaginary in our approximation) introduces a pi/2 phase shift, turning what was, say, a cosine in the E-field to a sine in the D-field. When I do (1/2)E.D, this results in something of the form sin(x)cos(x) (x being wave stuff) which boils down to something of the form sin(2x), which can of course be negative. Is all of this allowed? What has gone wrong if it isn't?

 

[mfb]

However, the complex relative permittivity (assumed totally imaginary in our approximation) introduces a pi/2 phase shift, turning what was, say, a cosine in the E-field to a sine in the D-field.

You are mixing a possible time-dependence and complex phases here. The phase difference between D and E is fixed and smaller than 90°, so the product will never be negative.

 

[jeffbarrington]

You are mixing a possible time-dependence and complex phases here. The phase difference between D and E is fixed and smaller than 90°, so the product will never be negative.

The product of two sinusoids less than 90 degrees out of phase can be negative though - what about sin(3.2415...)*sin(3.2415...-pi/4) or something like that? That is definitely negative.

You are mixing a possible time-dependence and complex phases here. The phase difference between D and E is fixed and smaller than 90°, so the product will never be negative.

To be sure, here's my working, I can't see the error. Also, the product of two sinusoids which are less than 90 degrees out of phase can certainly be negative, what about sin(3.2415...)*sin(3.2415...-pi/4) or something like that? Anyway, it's near enough exactly 90 degree phase difference here. Sorry if I'm missing something from your reply.

 

[mfb]

The product of two sinusoids less than 90 degrees out of phase can be negative though - what about sin(3.2415...)*sin(3.2415...-pi/4) or something like that? That is definitely negative.

This phase is not their complex phase! This can be phase in time or space, but this does not matter for your local energy density. E stays real, so D keeps its positive real part.

 

[jeffbarrington]

This phase is not their complex phase! This can be phase in time or space, but this does not matter for your local energy density. E stays real, so D keeps its positive real part.

I have a feeling this would only work if it were true that D = e_rcE, where D is complex and E is real, otherwise things mix together like I have done. If this were the case, I'd have D = contant*cos(pi/2)*E which seems fine. However, my notes claim that the E-field is also complex in the equation given.

 

[mfb]

Hmm... then energy density has to be something like the magnitude of the product, or the scalar product if you interpret the complex plane as two real axes (so D*E = |D|*|E|*cos(theta) with the angle theta between E and D).
Otherwise even a real e would lead to a negative energy density for imaginary E.

 

[PFfan01]

Homework Statement

The problem I have is that we are asked to show the complex relative permittivity of a good conductor is e_rc = 1 + i(sigma)/(omega*epsilon_0) where sigma is the conductivity and omega is the frequency of an electromagnetic wave in the medium. This is fine, I calculated it, the equation is given, it must be right. Now, we are also told that sigma is much greater than omega*epsilon_0, so the approximation it invites us to make is that e_rc = i(sigma)/(omega*epsilon_0). I think this is probably right so far.

The problem comes now. I am asked to find the energy density of the E-field in the wave. This is given by u = (1/2)E.D. Of course, this uses real values. I think D_c = e_rc*epsilon_0*E_c, and then I just take the real part of D_c to get D. However, the complex relative permittivity (assumed totally imaginary in our approximation) introduces a pi/2 phase shift, turning what was, say, a cosine in the E-field to a sine in the D-field. When I do (1/2)E.D, this results in something of the form sin(x)cos(x) (x being wave stuff) which boils down to something of the form sin(2x), which can of course be negative. Is all of this allowed? What has gone wrong if it isn't?

According to the theory for a plane wave in a moving medium, negative EM energy density can occur when the medium moves opposite to the wave vector at a faster-than-dielectric light speed. See: http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0167