Energy in electric field of an isolated particle? I learnt that the energy density of electric field is $\frac{\epsilon_0}2E^2$. However, I'm little confused about how can there be energy associated with an electron in empty space.
How can we have energy associated with this electron if there are no other charges around?
Thank very much.
 A: The work done to build a system is given by
$$U=\frac{\epsilon_0}2\int E^2dV$$
Thus applying this to an electron would mean that we are calculating the energy need to build an electron from many infinitesimal charges, $dq$.
This energy is called the self-energy of a particle and is due to its existence itself.
In classical theory one may imagine an electron to have many smaller parts, pushing and pulling on each other, whose interaction energy is given by the above mentioned integral.
Though if we solve this integral for the case of point charges we get infinity, which means that our model is not applicable to such particles. However, the concept of self energy is still valid for non point particle systems.
For example consider two systems, $\vec E_1$ and $\vec E_2$. Again the energy needed to build this system is
$$U=\frac{\epsilon_0}2\int E^2dV=\frac{\epsilon_0}2\int (\vec E_1+\vec E_2)^2dV$$
$$U=\frac{\epsilon_0}2\int E_1^2dV+\frac{\epsilon_0}2\int E_2^2dV+\epsilon_0\int \vec E_1\cdot\vec E_2dV$$
Here the first two integrals are the energy needed to build those systems respectively, their corresponding self energies. While the third integral is the energy due to their interaction with each other or the energy needed to build one system in the presence of the other.
