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Reference: "Theoretische Physik" (2015) by Bartelsmann and others, page 391, equation (11.23).

While deriving the first Maxwell equation based on Coulomb's law, the authors are using the following integral identity:

$$ \begin{align} \oint_{\partial V} \mathbf{d}f \cdot \mathbf{E}(\mathbf{r}) & = k\oint_{\partial V} \mathbf{d}f \cdot \int dV' \rho(\mathbf{r}') \frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \\ & = k \int dV' \rho(\mathbf{r}') \oint_{\partial V} \mathbf{d}f \cdot \frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3}, \end{align} $$

where $\mathbf{d}f$ is a surface element, $\mathbf{E}$ is electric field, $k$ is some constant, $\rho$ is volume specific charge density, $\mathbf{r}$ is location of reference charge, and $\mathbf{r}'$ is the integration variable.

Is it true that they first swapped the boundary and volume integral, and then pulled $\rho$ out of the boundary integral? Is it true that $\rho$ is a function of $\mathbf{r}'$, and can't be pulled out of the integral?


Since I could not follow the argument of the answer given by naturallyInconsistent, I made up an example and evaluated it once with integrals swapped, once left in the original order. Here is the result for $\rho = r$, evaluated at $\mathbf{r} = \mathbf{0}$, and integrated over a unit sphere:

Original order of integration:

$$ \Phi(\mathbf{r}) = \int_{\partial V} \left( \int_V \rho \frac{\mathbf{r} - \mathbf{r}'}{||\mathbf{r} - \mathbf{r}'||^3} \text{d}V \right) \cdot \mathbf{n} \text{d}A. $$


How not to do it

In what follows, the integration is done wrong. The misconception here is that the electric field $\mathbf{E}$ can be evaluated at a certain point $\mathbf{r}=\mathbf{0}$, and then integrated over the surface. But this would require the electric field to be constant everywhere. Instead, it is a function of position $\mathbf{E}(\mathbf{r})$.


First, compute $\mathbf{E}(\mathbf{0}):$ $$ \mathbf{E}(\mathbf{0}) = 8 \int_{\varphi=0}^{\pi/2} \int_{\theta=0}^{\pi/2} \int_{r=0}^1 \frac{r}{r^3} \left( \begin{array}{c} r \sin(\theta)\cos(\varphi) \\ r \sin(\theta)\sin(\varphi) \\ r \cos(\theta) \end{array} \right) r^2 \sin(\theta) \text{d}r \text{d}\theta \text{d}\varphi = \left( \begin{array}{c} 2 \pi/3 \\ 4 \pi/3 \\ \pi \end{array} \right) $$

Then, compute the flux:

$$ \Phi(\mathbf{0}) = 8 \int_{\varphi=0}^{\pi/2} \int_{\theta=0}^{\pi/2} \mathbf{E}(\mathbf{0}) \cdot \left( \begin{array}{c} \sin(\theta) \cos(\varphi) \\ \sin(\theta) sin(\varphi) \\ \cos(\theta) \end{array} \right) \sin(\theta) \text{d}\theta \text{d}\varphi = 2 \pi^2 + 16 \pi $$

Now if I swap the order of integration, I'm evaluating the expression

$$ \int_V \int_{\partial V} \left( \rho \frac{\mathbf{r}'}{||\mathbf{r}'||^3} \right) \cdot \mathbf{n} \text{d}A \text{d}V = \int_V \int_{\partial V} \left( \frac{\rho}{||\mathbf{r}'||^2} \right) \mathbf{n} \cdot \mathbf{n} \text{d}A \text{d}V = 8 \pi^2. $$

So there is at least one case where the order of integration may not be changed?

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    $\begingroup$ Boundary integrals and volume integrals can only be interchanged if the functions are nice, just like most integrals. But physicists don't generally care about these niceties, and will do it as long as it gives the right answer, which it generally does. $\endgroup$ Commented Jun 1, 2023 at 20:39
  • $\begingroup$ What is the question? $\endgroup$ Commented Jun 1, 2023 at 22:13

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The integral on the LHS only depends upon $\vec r$ whereas, after swapping the integrals, the $\vec r\,^\prime$ integral is on the outside, and so the $\rho$ that only depends upon $\vec r\,^\prime$ can be pulled out. The inner integral collects all the dependence upon $\vec r$

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  • $\begingroup$ Thanks a lot for your answer! I could not follow your argument, though, and made up an example which shows that there is at least one example where the order of integration may not be changed (given that the integration is done right). See the edit in the question. But it turns out that $\rho$ can actually be pulled out of the integral, as you pointed out. $\endgroup$ Commented Jun 2, 2023 at 15:06
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    $\begingroup$ Your evaluation of the electric field's integral is just wrong. It needs to be a function of position, but you got a constant vector. $\endgroup$ Commented Jun 2, 2023 at 16:54
  • $\begingroup$ The position is $\mathbf{r}'$. It is a function of the integration variables $(r, \theta, \varphi)$, and thereby not a constant. $\endgroup$ Commented Jun 2, 2023 at 19:17
  • $\begingroup$ Eventually, I got it, sorry. $\endgroup$ Commented Jun 3, 2023 at 8:14

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