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So we define the electrical energy density $(\bar U)$ between a parallel plate capacitor as $$ (\bar U) = \frac{1}{2} \epsilon_0 E^2$$ where $E$ is the electric field between the plates usually given by $$E = \frac{\sigma}{\epsilon_0}$$ My question is why do we define this physical quantity. As far as I can see there is no real practical utility for this. My teacher insists however there is a reason for us to define this and not just use energy, saying that if I were to derive the expression for the force on one of the plates then I would understand. I have done that as well but I am not able to see it. So please tell me what is the application of energy density.

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I'm sure there are many useful reasons.

But at the introductory level, this is usually derived assuming infinite parallel plates. So the total energy is infinite. Therefore, if you want to talk about the energy in this system it needs to be on a "per volume" basis.

Also, it turns out that this is true in general, and it gives us a way to determine the energy contained in the electric field of a system by computing $$U=\frac{\epsilon_0}{2}\int E^2\text dr^3$$

Also also, you end up getting a similar expression for magnetic fields, and these expressions are very important when considering energies associated with EM waves.

Typically teachers aren't out to screw you over by making up useless definitions. You can typically safely assume that if something is defined in your classes that it's because it has found good use in the larger physics community.

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Because it's useful later on, say, when you're in a more electrodynamical context, or when you're trying to formalise EM as a field theory do that you can quantize it. (Among many other reasons.)

But, frankly: if you're unwilling to entertain a definition when you cannot see an immediate use for them, then the only thing you're accomplishing is making it much harder for your future self to see interesting connections between concepts and subjects.

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  • $\begingroup$ Perhaps I should have been more clear but my teacher said that it has an immediate use that I could see right away (we have not done EM at all) $\endgroup$ – Prakhar Nagpal Apr 18 at 11:44
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    $\begingroup$ @Prakhar My point stands. $\endgroup$ – Emilio Pisanty Apr 18 at 11:45

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