# Electrostatic Energy Density

http://farside.ph.utexas.edu/teaching/em/lectures/node56.html

By equation 595, electric energy in the universe caused by an electric field is derived as http://farside.ph.utexas.edu/teaching/em/lectures/img1261.png. However, how come you can differentiate with respect to volume and say that the electric energy density is http://farside.ph.utexas.edu/teaching/em/lectures/img1262.png? After all, to come up with eq. 594, they had to make it a definite integral over all space so that the left term (in eq. 593) would converge to 0, and in order to do say the latter, wouldn't it have to be an indefinite integral?
Analogously, the integral from negative infinity to infinity of x/(x4 + 1) is 0, but you cannot say that the function is equal to 0 for all x.
-Jim

PS: sorry i don't know how to use LaTeX and this post is kind of unclear

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## Answers and Replies

I think you are right.

I checked in an old textbook, which says that 595 is a (reasonable) definition that can't be derived from integral equations like 594.

It also reports that is a questioned definition. Somebody think that try to define an energy density of an electrostatic field is like try to define a density of beautifulness in a painting masterpiece.

Dale
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I don't really understand the confusion here. Whenever you have an integral over some volume the integrand is by definition a density. If the result of the integral has units of mass then the integrand has units of mass/volume and is a mass density. If the result of the integral has units of energy then the integrand has units of energy/volume and is an energy density.

May be the confusione is due to the fact that that density function is not unique, you can just add any function whose integral is 0 on V. Among all those functions, 595 has been choosen, by definition, as energy density.

More subtle arguments could be needed in the discussion on the very possibility to define an energy density as a physical entity in this case.

you can just add any function whose integral is 0 on V

Its true and we can consider that function as some kind of constant. But it is useless and just makes our problem more complicated.

from CE book (3rd edition - Jackson):
"The expression for energy density is intuitively reasonable, since regions of high fields 'must' contain considerable energy."
I hope this quote is enough to define energy density as a physical entity.

It is what I intended when I said "(reasonable) definition" in my first post.

I'm not able to enter in the discussion about the physical entity. Probably, it is related to the question on where the energy is stored: in the field or in the charges. In General Relativity it seems that it is necessary to locate it in the field.

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Attached is a page from the Feynman Lectures on Physics, where this issue is briefly discussed. The ambiguity that exists in the definition of energy density is much greater than the ambiguity that exists in energy expressions more generally. In classical mechanics, energy is defined only up to an additive constant, so you're free to choose what to consider the zero level of the potential energy. (My understanding is that special relativity, since it yields a formula for the rest energy of a particle, eliminates at least this ambiguity, but I could be wrong. If I am, someone please correct me.) However, even if you allow an additive constant, the energy density in classical electromagnetism is still underspecified. As someone mentioned earlier general relativity should provide a way to find the correct expression for energy density, but unfortunately the effects of classical electromagnetism and the effects of general relativity manifest themselves in vastly different length scales.

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