Question

Let us model an electron not as a point charge, but as a sphere of radius R over whose

surface the the charge −1.6 × 10−19 C is distributed uniformly.

(a) Calculate the energy density u(r) of the electric field at all points in space (i.e.,

both inside and outside the sphere).

(b) Calculate the total energy associated with the electron’s field. There are at least two equivalent ways to do this: (1) You can integrate the energy density directly, summing the energy in each tiny region of space. This will require an integration, since the energy density varies from point to point. To carry this out, divide space into thin spherical shells centered on the electron, of (inner) radius r and thickness dr. For small enough dr – and it will get small enough since we will let it go to zero! – the energy density is essentially constant over the shell. In addition, for small dr the volume of the shell is dV = 4πr2dr, and so the energy contained in such a shell is

dU = u(r)dV = u(r)4πr2dr

Now sum over the shells by integrating this over some appropriate range in r. The result is the total field energy in all space. (2) Treat the shell as a capacitor relative to infinity (i.e., with the second conductor located at infinity). Calculate the potential difference between the shell and infinity, and use U = (1/2)QV .

(c) According to the theory of relativity, energy has mass (inertia): m = E/c2. So the energy in the field is part of the mass of the electron! Assume the entire mass of the electron is due to the energy of its electric field, and calculate a numerical value for R. This is known as the classical radius of the electron.

Answer 1

(a) The electric field inside the sphere will be zero as the whole charge -1.6x 10^-19 C is distributed over the surface. To find the electric field outside the sphere, we will use Gauss's law which says that

Choose a Gaussian surface with radius r > R . Then

where e = 1.6 x 10^-19 C

Now the Energy density u(r) is given as

(b) We will follow the first approach.

dU= u(r)dV

where dV =4 pi r²dr

(c) E=U

E= mc² = U