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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?

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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.

This argument then gives a particularly clean, fancy-integral-free path to the calculation of the interaction cross term in the energy of the electric fields of two point charges.

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