# Spherical conductors

Gold Member

## Homework Statement

Given two spherical conductors of radius R and tangent at O, both are charged and in equilibrium with surface charge density ρs0cos2theta. Calculate:
a) Voltage of both spheres at O. (SOLUTION: V=2ρ0R/(3ε0)
(...)

## 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 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,
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

## Answers and Replies

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
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
Science Advisor
Homework Helper
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 :) )

Gold Member
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 cant calculate ds for the integral

#### Attachments

• drawing.png
3.8 KB · Views: 378
haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
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 cant 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?