Purcell 1.77 Electron jelly

[Rob2024]

Homework Statement

Imagine a sphere of radius a filled with negative charge of uniform
density, the total charge being equivalent to that of two electrons.
Imbed in this jelly of negative charge two protons, and assume that,
in spite of their presence, the negative charge distribution remains
uniform. Where must the protons be located so that the force on
each of them is zero? (This is a surprisingly realistic caricature of
a hydrogen molecule; the magic that keeps the electron cloud in
the molecule from collapsing around the protons is explained by
quantum mechanics!)

Relevant Equations

$U= \frac{k Qq}{r}, E = \frac{kQ}{r^2}$

I initially tried energy method, I realized this cannot be minimized.

$U = \frac{kq^2}{2r} - \frac{2 k qq'}{r} , q' = \frac{2e r^3}{R^3},q = e$
$U = \frac{kq^2}{2r} - \frac{2 k q\frac{2e r^3}{R^3}}{r}$
$= k q^2 ( 1/2r -4 r^2/R^3)$
$= \frac{1}{2} k q^2 ( r - 8 r^2/R^3)$
$U' \sim -1/2r^2 - 8 r/R^3 = 0$

The force balance method works correctly.

$E = \frac{k q}{4 r^2}, E' = \frac{k q'}{r^2} = \frac{ k 2 q r^3/R^3}{r^2}$
$r = \frac{1}{2} R$


I cannot figure out what is causing this inconsistency.

 

[Steve4Physics]

The meanings of your symbols are not always clear and you have not provided any explanations of the rationale. But I think I can guess. Also you have used '$R$' for the sphere's radius but the question calls it 'a'.

Consider the potential a distance $r$ from the centre of the sphere due to the negative charge alone. The potential is not $\frac{k \frac{-2e r^3}{R^3}}{r}$.

You can try working out the correct expression for yourself or do a search such as 'potential inside a uniformly charged sphere'.

 

[Rob2024]

Thanks, this worked. I used the potential energy incorrectly.