Spherical Conductors: Voltage Calculation & Equilibrium | ρs=ρ0cos2theta

[pitbull]

Homework Statement

Given two spherical conductors of radius R and tangent at O, both are charged and in equilibrium with surface charge density ρ_s=ρ₀cos²theta. Calculate:
a) Voltage of both spheres at O. (SOLUTION: V=2ρ₀R/(3ε₀)
(...)

Homework Equations

The Attempt at a Solution

So I tried to solve it, first saying that both have the same voltage on its surface, thus, the voltage at point O is the same as voltage anywhere else on the surface of any of those spheres. I integrated ρ_s on the surface on one sphere, (R between 0 and R, and theta between 0 and 2pi), and I got a charge of R²ρ₀pi/4 on one sphere,
Then I use Gauss to find the Electric field made by such sphere and integrate to find the voltage on the surface of the sphere, and I got V=ρ₀R/(8ε₀). I cannot find what's wrong

 

[haruspex]

Where theta is...?
I'm a bit puzzled, though. In principle, one could deduce the surface charge distribution from the total charge and other information. Are we to suppose that the given formula is the solution? Or is 'conducting' a mistake here?

 

[BvU]

I integrated ρs on the surface on one sphere, (R between 0 and R, and theta between 0 and 2pi), and I got a charge of R2ρ0pi/4 on one sphere,

Can you show what you did ? A sphere sounds three-dimensional. What happened to $\phi$ and why do you let $\theta$ go from 0 to $2\pi$ ? What do you think the charge density at O is ?

A drawing might make things a lot clearer, also for potential helpers (see the confusion with haru, who will help you further, since it's past my bedtime here :) )

 

[pitbull]

Where theta is...?
I'm a bit puzzled, though. In principle, one could deduce the surface charge distribution from the total charge and other information. Are we to suppose that the given formula is the solution? Or is 'conducting' a mistake here?

Can you show what you did ? A sphere sounds three-dimensional. What happened to $\phi$ and why do you let $\theta$ go from 0 to $2\pi$ ? What do you think the charge density at O is ?

A drawing might make things a lot clearer, also for potential helpers (see the confusion with haru, who will help you further, since it's past my bedtime here :) )

The solution is not a formula, it is just what the textbook gives as a solution. I don't know why I was thinking of polar coordinates, so I have the wrong limits for the integral . I just saw the drawing. It was on the back of the page, so now it should make sense. But now that I saw the drawing, I can't calculate ds for the integral

 

[haruspex]

The solution is not a formula, it is just what the textbook gives as a solution. I don't know why I was thinking of polar coordinates, so I have the wrong limits for the integral . I just saw the drawing. It was on the back of the page, so now it should make sense. But now that I saw the drawing, I can't calculate ds for the integral

OK, the diagram helps. With theta defined that way, I confirm that the formula for charge density is indeed a solution.
But it's easier to work in terms of angle subtended at the centre of the sphere. Let A be the point where the chord shown touches the sphere at top right. What angle does the chord OA subtend at the centre of the sphere? Call this angle $\phi$. Consider the circular band width $d\phi$ passing through A. What potential does that produce at O?