The maxwell stress tensor has an important feature at the end of the equation, given as: \(\frac{1}{2}(\epsilon E^2 + \frac{1}{2 \mu}B^2) \delta_{ij}\) What is this quantity I see? Is it the total electric and magnetic density of the field?
You want to know what the \(\delta_{ij}\) is? Do you even know what a tensor is? That's a pretty important if you're trying to understand the stress energy tensor. Staring straight ahead and ignoring the above, \(\delta_{ij}\) is zero when \(i \neq j\) and 1 when \(i = j\). i and j are in this case spacetime indices, for example, you can extract the energy density from the stress tensor by looking at it's tt component, so straight away you can see that all the non zero entries of this stress tensor have units of energy. Anyway, without this term (that comes with a minus sign) you'd be overcounting the diagonal components because there is a term when i = j and another (identical) one when j = i. This term deals with that.
For goodness sake, I know what that is. I'm asking for the entire expression sir, not the function at the end. Of course it's incomplete, it was taken from maxwells tensor. Get it right - i asked what the entire expression resembled.
I see the field density in there, I also see the magnetic field density - Is this (JUST LIKE I ASKED) the total electric and magnetic field densities?
The electric field, for instance, has a density of \(1/2 \epsilon. E^2\) where E is the electric field. The other collection of terms make up the magnetic density of the field, do they not?
No, that is the energy of the electric field, and the other term is the energy of the magnetic field. It's not at all surprising when by definition the stress tensor contains the energy of the fields is it?
I think you are wrong sir ~ http://en.wikipedia.org/wiki/Electric_field The wiki page explains the terms given: \(u_e=\frac{1}{2} \epsilon E^2\) where ε is the permittivity of the medium in which the field exists, and E is the electric field vector. It's written on that page.
More specifically, its entire form describes the energy-density of the field - the term E^2 as I said, is the electric field, not the energy per se.
The point is that the stress tensor, and more specifically the tt component of the stress tensor, gives the energy density of the fields. I wrote that in my first post on this thread. It's not at all surprising that expressions for the energy density of the various fields crop up in the stress tensor. Anyway, it makes a lot more sense (to me at least) to write the stress tensor in terms of the field strength rather than the component fields because the stress tensor is (more or less) the variation of the Lagrangian with respect to the metric, and the field strength is what appears in the Lagrangian.
I have a feeling that was you saying ''ok, so be it.'' Now we understand each other, is the expression the total densities of the electric and magnetic field? I would imagine it is...
I'm going to write this one more time just for the hell of it: NO! The expression you wrote, if we ignore the fact that it has tensor structure, is the energy density of the 2 fields. The "total density" of the electric and magnetic fields is a phrase you made up that has no meaning other than in your own tiny brain. Also, the stress tensor is far more than just the energy density of the fields, which is the tt component thereof and the expression you write for the stress tensor has to deal with that. The term you are talking about comes with a minus sign in the expression for the stress tensor which is pretty important if you want to call it an energy density. I have a feeling that you will continue to be a moron about this so my contribution such as it is will be over, but please do prove me wrong - firstly you could actually learn what a tensor is.
And eh, by the way... I know the tensor is much more elaborate than a simple expression for the field densities. Well, you weren't much help, maybe... until the end. It turns out that the expression contained within the paranthesis is in fact the volumetric energy density of the electric field. See, I do try without the calaboration here: ''Electric and magnetic fields store energy. In a vacuum, the (volumetric) energy density (in SI units) is given by'' http://en.wikipedia.org/wiki/Energy_density#Energy_density_of_electric_and_magnetic_fields Thanks for the overwhelming help sir.
That's what prometheus was saying, though. The \(E^2\) term is the energy density contribution from the EM field's electric component, the \(B^2\) term is the analogous contribution from the magnetic component.
I said that the quantity was the electric field density where E is the electric field. He said a firm ''No'' to this, saying it was the energy of the field.
Is it just me? I see a difference in the two. Not to mention Prometheus did not have a clue of the term ''field density'' ~ which is the only description I've ever heard for the quantity.
Well, they're usually talking about field's energy density when they say that. After all, what other densities are there to talk about? What units would it have? There are electric charge and magnetic current densities, but those aren't field densities. My recommendation: try to look these things up on Wikipedia from the outset, it usually helps clarify confusions or at least ensures you have some relevant background (or tried to obtain some) before coming here to ask people about it.
