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This question is from Griffiths' Electrodynamics, Chapter 2, question 49, part e). The question is as follows:

Problem 2.49 Imagine that new and extraordinarily precise measurements have revealed an error in Coulomb's law. The actual force of interaction between two point charges is found to be $$\mathbf{F}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\left(1+\frac{r}{\lambda}\right)e^{-r/\lambda\hat{r}}$$ where $\lambda$ is a new constant of nature (it has the dimensions of length, obviously, and is a huge number - say half the radius of the known universe - so that the correction is small, which is why no one ever noticed the discrepancy before). You are charged with the task of reformulating electrostatics to accomodate the new discovery. Assume the principle of superposition still holds.
(a) What is the electric field of a charge distribution $\rho$ (replacing Eq. 2.8)?
(b) Does this electric field admit a scalar potential? Explain briefly how you reached your conclusion. (No formal proof necessary - just a persuasive argument.)
(c) Find the potential of a point charge $q$ - the analog to Eq 2.26. (If your answer to (b) was "no", better go back and change it!) Use $\infty$ as your reference point.
(d) For a point charge $q$ at the origin, show that $$\oint_{\mathcal{S}}\mathbf{E}\cdot d\mathbf{a}+\frac{1}{\lambda^2}\int_{\mathcal{V}}Vd\tau=\frac{1}{\epsilon_0}q,$$ where $\mathcal{S}$ is the surface, $\mathcal{V}$ the volume, of any sphere centered at $q$.
(e) Show that this result generalizes: $$\oint_{\mathcal{S}}\mathbf{E}\cdot d\mathbf{a}+\frac{1}{\lambda^2}\int_{\mathcal{V}}Vd\tau=\frac{1}{\epsilon_0}Q_{\text{enc}},$$ for any charge distribution. (This is the next best thing to Gauss' law in the new "electrodynamics".)

So I did part d) which is for a point charge at origin. Now to solve part e), can I just assume that I have $n$ charges in a given volume $V$ with surface area $S$ and then take the summation on both sides of d) as $i$ goes from $1$ to $n$? The thing that motivates me to do so is since the electric field follows superposition principle, I can sum the electric field due to all the charges over the surface and similarly since the potential is a scalar, I can use the principle of superposition as well. Is this a rigorous way to prove it?

Editors' Note: the $r$ in equation one should actually be cursive; I am unsure how to make it look right.

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  • $\begingroup$ Assume that the charge enclosed is made up of infinitely many smaller charged particles. Since you are going to work on a closed surface, use solid angles while finding the flux. $\endgroup$ – Yashas Feb 14 '17 at 4:44
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First thing: you realize there is a full solution manual for this textbook right? The solution there is perfectly valid.

Or maybe you don't like the way the solution manual solved, or maybe you are just curious if your method works, does it?

No. It does not unfortunately. To see this realize that what you did on item d) was just proving the "new electrodynamics Gauss Law" (NEGL) for the particular case of a point charge at the center of the sphere. If you have a distribution of charge on item e) they will not be all at the center of the sphere, so there is no way to "sum both sides" and get away with it.

You could in principle generalize your result in d) for a charge place anywhere inside the sphere (but then is not your original idea), and this might give you an edge on proving e) for the particular case of a SPHERE surrounding a distribution of charge, you still wouldn't be able to prove it for any shape. To do this, the method that the solution manual presents is as simple as it can possible be.

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