Consider a simple but surprisingly accurate model for the hydrogen molecule: two positive point charges, each having charge e, are placed inside a uniformly charged sphere of radius R, which has a charge equal to -2e. The two point charges are placed symmetrically, equidistant from the center of the sphere. Find the distance from the center, a, where the net force on either point charge is zero. (Use the following as necessary: k, R, and e.)

a = R/2

Explanation:

Let R be the radius of the sphere and q = -2e be the charge in it. Let q₁ be the charge at radius a where the one of the point positive charges e is located. . Since the charge is uniformly distributed, the volume charge density is constant. So, q/4πR³ = q₁/4πa³

q₁ = q(a/R)³ = -2e(a/R)³. The electric force due to q₁ at r is F₁ = kq₁q/a² = kq²(a/R)³/a² = k(-2e)²a/R³ = 4ke²a/R³.

Let h be the distance between the two point charges. The electric force due to the one point charge on the other is F₂ = ke²/h²

If the net force on either charge is zero, then

F₁ = F₂

4ke²a/R³ = ke²/h²

a = R³/4h²

Since h = 2a, since the charges are equidistant from each other,

a = R³/4(2a)² = R³/8a²

a = R³/8a²

a³ = R³/8

a = ∛(R³/8)

a = R/2