Well, I did the problem. The problem is I KNOW my answer is wrong. Anybody have any idea why? (This is what I've worked out with my friend).(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

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!)

[Ahhhh logically when I try to think about this, I don't understand how there's ANY way there could be zero force on the protons in a sphere of negative electric charge. But that's non-mathematical.]

2. Relevant equations

Gauss's Law (sphere)

[tex] E = \frac{Q}{r^2} [/tex]

q = e (charge of an electron)

a = radius of sphere

r = radius of Gaussian sphere

3. The attempt at a solution

So, supposedly, I'm supposed to take a Gaussian surface/sphere within the sphere on which the charges will rest.

The charge density of the sphere is

[tex]\frac{-2q}{\frac{4}{3}\pi a^3}[/tex]

or [tex]\frac{-3q}{2\pi a^3}[/tex].

Therefore the charge on my Gaussian sphere is the charge density times the volume of the Gaussian sphere, or

[tex](\frac{-3q}{2\pi a^3})(\frac{4}{3}\pi r^3)[/tex]

[tex]=\frac{2qr^3}{a^3}[/tex]

[tex]E = q_total/r^2[/tex]

Total charge is 2 electrons - charge inside Gaussian sphere

[tex]q_{total} = 2q - \frac{2qr^3}{a^3}[/tex]

[tex]E = \frac{1}{r^2}(2q - \frac{2qr^3}{a^3}[/tex]

[tex]E = \frac{2q}{r^2} - \frac{2qr}{a^3}[/tex]

When the E field is equal to zero, there is no force.

[tex]0 = \frac{2q}{r^2} - \frac{2qr}{a^3}[/tex]

[tex] \frac{2q}{r^2} = \frac{2qr}{a^3} [/tex]

[tex]r^3 = a^3[/tex]

[tex]r = a[/tex]

The more I look at this the more it doesn't make sense, but I can't pinpoint where the "wrong" is.

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# Homework Help: Electric Field Forces in a Hydrogen Molecule

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