Why does this integral calculating the electrostatic energy converge?

Question

I came across the problem of calculating the interaction energy of two point charges separated by some distance a in Griffith's Introduction to Electrodynamics. Here and everywhere else that I look, I find that the calculation is done by using the expression for interaction energy: $$\epsilon_{0}\int\vec{E}_{1}\cdot\vec{E}_{2}\,d\tau$$ where the integration is done on all space.

Why does this give correct result when $\vec{E}_{1}$ and $\vec{E}_{2}$ are both undefined at the points where the two charges are situated? How come does the integral comes out as finite when the integrand goes to infinity at two places in the domain of integration?

Answer 1

This converges because the integral can be thought of as finite. To put it more rigorously, let's look at the integral of two vector functions $\vec{v}_1(\vec{r})$ and $\vec{v}_2(\vec{r})$ near the origin, where $\vec{v}_1$ is bounded and $$ \vec{v}_2 = \frac{1}{r^2} \vec{w}(\vec{r}), $$ where $\vec{w}(\vec{r})$ is bounded.

Now, the integral we want to perform is $$ I = \iiint_\text{all space} \vec{v}_1 \cdot \vec{v}_2 \, d\tau $$ but we are worried that the integrand diverges as $r \to 0$. We can "isolate" the problematic region by splitting up the integral: $$ I = \iiint_{|\vec{r}| \geq \epsilon} \vec{v}_1 \cdot \vec{v}_2 \, d\tau + \iiint_{|\vec{r}| \leq \epsilon} \vec{v}_1 \cdot \vec{v}_2 \, d\tau. $$ We'll assume that the first integral is well-defined and look more closely at the second. If we look at this carefully, we can put a bound on the integral into a ball of radius $\epsilon$ around the origin and the rest of space: \begin{align*} \left| \iiint_{|\vec{r}| \leq \epsilon} \vec{v}_1 \cdot \vec{v}_2 \, d\tau \right| &= \left| \iiint_{|\vec{r}| \leq \epsilon} \frac{\vec{v}_1 \cdot \vec{w}}{r^2} \, d\tau \right| \\ &\leq 4 \pi \max_{|\vec{r}| \leq \epsilon} |\vec{v}_1 \cdot \vec{w}| \int_0^{\epsilon} \frac{1}{r^2} r^2 \, dr \\ &= 4 \pi \max_{|\vec{r}| \leq \epsilon} |\vec{v}_1 \cdot \vec{w}| \epsilon. \end{align*} So we can see that in the limit as $\epsilon \to 0$, the contribution from this portion of the integral becomes negligible. This means that it is well-defined to talk about the "integral over all space" as the limit as $\epsilon \to 0$ of "the integral over all space except for small spheres of radius $\epsilon$ around points where $\vec{v}_2$ diverges."¹

In a real sense, this works because the singularity of $\vec{v}_2$ is sufficiently "weak" that we can still get a sensible limit out of this. This happened because the singularity was proportional to $1/r^2$, which was cancelled out by the volume factor of $r^2$ in the integral. If you applied the same logic to the quantity $\vec{v}_2 \cdot \vec{v}_2$ near the origin, the integrand would be proportional to $1/r^4$ near the origin, which is "too strong" of a singularity to be cancelled out by the volume factor. This is why Griffiths has to discuss the charge's "self-energy" separately from the "interaction energy."

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¹ To really be rigorous about this (and avoid doing a suspect $0/0$ cancellation in the demonstration above), we would instead want to fix some radius $r_0$ and take a sequence of $\epsilon$ values $\epsilon_1, \epsilon_2, \dots$ such that $\lim_{n \to \infty} \epsilon_n = 0$. We can then show, using similar logic, that the sequence of integrals $$ I_n \equiv \iiint_{\epsilon_n \leq |\vec{r}| \leq r_0} \vec{v}_1 \cdot \vec{v}_2 \, d\tau $$ also converge to some finite limit. This limit would then be what we mean by "the integral over the sphere of radius $r_0$ centered at the origin." The logic is much the same, but somewhat more opaque a first pass, which is why I phrased my main answer the way I did.

Answer 2

$|E|$ blows up as $r^{-2}$ but the measure in spherical coordinates centered about one of the charges the integration measure is $d^3x= r^2 \sin\theta\,dr\, d\theta\, d\phi$, so the $r^2$'s cancel to give a finite integrand.

Answer 3

This is a way to prove convergence and also to understand why this integral represents potential energy between two charge distributions. Another way to look at this is

$$\iiint \frac{1}{2} \epsilon_0(\vec{E}_1+\vec{E}_2)^2d\tau $$ : $$\iiint \frac{1}{2} \epsilon_0(\vec{E}_1)^2d\tau +$$

$$\iiint \frac{1}{2} \epsilon_0(\vec{E}_2)^2d\tau +$$

$$\iiint \epsilon_0\vec{E}_1\cdot\vec{E}_2\,d\tau$$ The last component represents PD between charge distributions, $$=\iiint \epsilon_0\vec{E}_1\cdot(-\nabla V_{2})\,d\tau.$$ Rearranging the vector field identity $$\nabla \cdot (V_{2}\vec{E}_1) = V_{2} \nabla \cdot \vec{E}_1 + \vec{E}_{1} \cdot \nabla V_{2},$$ gives $$\vec{E}_{1} \cdot (-\nabla V_{2})= -\nabla\cdot(V_{2}\vec{E}_1) + V_{2} \nabla \cdot\vec{E}_1.$$ Substituting this in the integral gives $$\iiint\epsilon_0\left[-\nabla \cdot (V_{2}\vec{E}_1) + V_{2} \nabla \cdot \vec{E}_1\right]d\tau$$ $$=-\iiint\epsilon_0 \nabla \cdot (V_{2}\vec{E}_1)\,d\tau+\iiint \epsilon_0V_{2} \nabla \cdot \vec{E}_{1}\,d\tau.$$

Invoking Stokes' Theorem on the first integral gives $$-\iint\epsilon_0 (V_{2}\vec{E}_1) \cdot d\vec{S}+ \iiint\epsilon_0 V_{2} \nabla \cdot \vec{E}_1\,d\tau.$$ With the integration volume as all of space, the surface integral evaluates to zero for localized sources, since at large distances $(r\rightarrow\infty)$, the integrand falls off as $V_{2}\vec{E}_{1}\sim r^{-3}$. This leaves $$\iiint\epsilon_0V_{2} \nabla \cdot \vec{E}_1\,d\tau.$$ Invoking Gauss' Law, $$\iiint\epsilon_0 V_{2} \frac{\rho_1}{\epsilon_0}\,d\tau$$ $$=\iiint V_{2} \rho_1\,d\tau$$

In this form, it is clear to see why it represents potential energy, as you are building up charge distribution 1 in the presence of potential 2.

In this form you can also understand why it converges, as the $V\rho$ is always a finite number, because $V$ is never evaluated at any point that causes it to be infinity.