Problem understanding energy of a electric field

Question

I just finished classical mechanics, and waves and have started electromagnetism. I have studied upto electric fields, potentials and conductors in a static field and have just entered capacitors. And I just read what some books call the energy of a electric field. I'm having a pretty hard time understanding it.

My conception about energy is that the quantity $\frac12m\dot r^2$ of a particle is called it's kinetic energy and when in the field of an conservative force the quantity $\int_A^B \vec F .d\vec r $ is called the change ($U_A-U_B)$ in a potential energy function. the quantity $$\frac12m\dot r^2+\sum U_i(\vec r) $$ is constant in time until there is some work done by some external force.

so I don't understand how the field has energy or how is it defined or anything. I don't really believe that energy is the "measure of how energtic a man is" or "how energy is not 'destroyed ever'" I simply believe it's just a number that helps us keep track of motion of a system. Please explain with some undergraduate and good mathamatics. I'm sorry that I'm stupid

Answer 1

It might help to think of the energy of the electric field as a kind of potential energy. If you bring two like charges together, that takes energy. They're trying to repel each other. Thus, once they're brought together, they have a large amount of potential energy. One way to account for that energy is to say each particle has some potential energy. Another way to account for that energy is to say the particle's don't have potential energy; rather, the electric field has potential energy. The particles have "stored" energy in the field.

This might seem like an odd way to think about energy when you're just doing electrostatics. But this turns out to generalize nicely to things like electromagnetic waves, so it's useful to get acquainted with the idea now.

Answer 2

Field is not any abstract fairy-tale 'mathematical fiction' concept; it has momentum and energy density; it exchanges momentum and energy with charged entities.

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Energy conservation is a local process which evidently implies electromagnetic field between two interacting charges must mediate the energy and momentum exchange between the charges and hence must have energy density and momentum.

Let the electromagnetic field has $u$ as its energy density (amount of energy per unit volume in the field) and let $\bf S$ represents the energy flux- the amount of energy per unit time flowing across a unit area perpendicular to the flow).

Electromagnetic field, as said at the outset, can interact with matter and do work on them; so this energy interaction must be considered when discussing energy conservation.

The field energy inside a volume $V$ is $\displaystyle \int_V u\,\mathrm dV\;.$ Amount of energy flowing out of the volume $V$ is given by $\displaystyle \int_\Sigma \mathbf S\cdot n\,\mathrm da \;.$

Now, the work done per unit time by the field on the matter inside the volume $V$ is given by $\displaystyle \int _V Nq(\mathbf E + \mathbf v\times \mathbf B)\cdot \mathbf v\,\mathrm dV$ where $N$ is the number of particles per unit volume; this can be written as \begin{align}\int _V Nq(\mathbf E + v\times \mathbf B)\,\mathrm dV& = \int_V Nq\mathbf E\cdot v\,\mathrm dV\\ &= \int_V \mathbf E\cdot \underbrace{(Nq\mathbf v)}_\textrm{current-density}\,\mathrm dV\\ &= \int_V E\cdot \mathbf J\,\mathrm dV \end{align}

So, the continuity equation is written thus: $$\underbrace{-\frac{\partial}{\partial t}\int_V u\,\mathrm dV}_\textrm{rate of change of energy inside volume $V$}=\underbrace{\int_\Sigma \mathbf S\cdot \mathbf n\,\mathrm da}_\textrm{amount of $\mathbf{field\, energy}$ flowing out of volume $V$ per unit time} + \underbrace{\int_V \mathbf E\cdot \mathbf J\,\mathrm dV}_\textrm{work done per unit time by the field on the matter inside volume $V$} \;. $$

After some algebra, it can be inferred that

$$u = \frac{\epsilon_0}2 \,\mathbf E\cdot \mathbf E+\frac{\epsilon_oc^2}{2}\,\mathbf B\cdot \mathbf B\;.$$

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It boils down to much simpler interpretation in the case of statics as aired by Jahan Claes.

The energy $U$ is the potential energy associated with configuration of the charges of the system.

If you keep three charges very, very far from each other, then the potential energy of the system is very effectively zero.

If you keep three charges very, very far from each other, then the potential energy of the system is very effectively zero.

But when you bring them close together to a specified coordinate, then the potential energy of the system increases from $0$ to a certain positive value $U\;.$

For a discrete system of $N$ charges, the potential energy associated with their configuration is given by

\begin{align}U&= \frac12\,\sum_{j=1}^N \, q_j\sum_{k\ne j}\,\frac1{4\pi\epsilon_0}\,\cdot \frac{q_k}{r_{jk}}\\ &= \frac12\,\sum_{j=1}^N \, q_j\,\varphi(\mathbf r_j) \end{align}

For a system of continuous charges, $$U= \frac12\int \rho(\mathbf r)\,\varphi(\mathbf r)\,\mathrm d^3\mathbf r \;.$$

After some manipulation, it can again be inferred that

$$U= \frac{\epsilon_0}2 \int\,\mathbf E\cdot \mathbf E\,\mathrm d^3\mathbf r\;.$$

This is the energy required to assemble the system of charges in that specific configuration.