# Why does the interaction energy for the electric fields of two point charges vanish over any shell smaller than their separation?

A previous question, A triple integral in Spherical coordinates from Jackson's book on Electrodynamics, presents the fairly routine calculation of the interaction energy encoded in the cross-term of the electric fields produced by two point charges, $$E=\iiint \frac {(\vec x - \vec x_1)\cdot(\vec x- \vec x_2)}{|\vec x - \vec x_1|^3 \,|\vec x- \vec x_2|^3} \mathrm d^3 \vec x .$$ As shown in that thread, this integral can be fairly cleanly expressed in polar coordinates centered on the charge at $$\vec x_1$$ and with the polar angle $$\theta$$ measured from a $$z$$ axis placed along the $$\vec x_1$$-$$\vec x_2$$ axis, in which case it reads $$E = 2\pi \int_0^\infty\!\!\! \int_0^\pi \frac{r-d\cos(\theta)}{\left(r^2+d^2 -2rd\cos(\theta)\right)^{3/2}} \sin(\theta) \,\mathrm d\theta \,\mathrm dr ,$$ with $$d$$ the distance between the two charges.

Now, one very curious thing that happens within this routine calculation is that the internal integral over the angle, $$E (r) = 2\pi \int_0^\pi \frac{r-d\cos(\theta)}{\left(r^2+d^2 -2rd\cos(\theta)\right)^{3/2}} \sin(\theta) \,\mathrm d\theta ,$$ is identically equal to zero for all $$r$$ between $$0$$ and the inter-charge distance $$d$$, coming from a fairly precise cancellation of the various contributions at different polar angles $$\theta$$.

For a slightly closer look, here is the polar integrand as a function of $$\theta$$ (horizontal axis) and $$r$$ (vertical axis), with negative integrands in blue and positive integrands in red. The integral over any horizontal strip is zero, but while at low $$r$$ the integral is essentially sinusoidal, as it goes from zero to closer to $$d$$ from below, the cancellations are increasingly between a broad, low band of positive integrand and a short, narrow, increasingly singular band of negative values at low $$\theta$$.

With all that said: Is there a clean physical interpretation for this property of this integral? It looks too exact to be a coincidence. Can it be derived a through more direct, less mathematical route which also imbues it with physical significance?

And similarly, while we're here: without giving away the precise answer to a homework question, when $$r>d$$ the angular integral has a particularly clean power-of-$$r$$ form, $$E(r)\propto r^{-2}$$, which also seems like it should have a clean physical interpretation. Is that the case?

The result follows cleanly and simply from the integral form of Gauss's law when suitably interpreted: the angular integral, once rephrased as $$E(r) = \iint_{S^2(r)} \frac{\vec x-\vec x_2}{|\vec x-\vec x_2|^3}\cdot \hat{n} \, \mathrm d\Omega$$ with the electric field from the charge at $$\vec x_1$$ replaced by the radial unit normal $$\hat{n}$$, is simply the flux of the electric field from the charge at $$\vec x_2$$ over a the sphere $$S^2(r)$$ of radius $$r$$; Gauss's law then guarantees that this will vanish if the charge at $$\vec x_2$$ is outside of the shell, and it also gives it a constant value if it's inside the shell.