# Charged Sphere sliced, force required to keep them as they were

1. Jul 19, 2012

### AGNuke

A metallic sphere of radius R is cut in two parts along a plane whose minimum distance from the sphere's centre is h and the sphere is uniformly charged by a total electric charge Q. What minimum force is necessary to hold the two parts of the sphere together?

The Real trivia
The solution which we were "encouraged" to come up was using a term called "electric pressure", defined as the electrostatic force per unit area. By multiplying it with the base area of the cross-section obtained after slicing the sphere, I got the answer.

$$P_{el}=\frac{\sigma ^{2}}{2\varepsilon _{0}}; \; \sigma =\frac{Q}{4\pi R^2}$$
$$F_{el}=P_{el}\times S; \; S=\pi(R^2-h^2)$$
$$F_{el}=\frac{\frac{Q^2}{16\pi^2R^4}}{2\varepsilon_{0}}\times \pi(R^2-h^2)=\frac{Q^2(R^2-h^2)}{32\pi\varepsilon_{0}R^4}$$

Now I actually didn't get the concept here, what was that supposed to mean. Isn't there a conventional way to solve this problem?

2. Jul 19, 2012

### TSny

Hi AGNuke. I'm not too clear on what specific question you are asking. Are you asking why the pressure is given by σ2/2εo or are you asking why the force is given by F = P*$\pi$(R2-h2)? Or are you asking something else?

The expression for the pressure can be obtained in a fairly conventional way by considering the force on a small patch of area of the surface of a charged conductor.

3. Jul 19, 2012

### AGNuke

TSny, I am sorry if I was unable to convey my question properly, but I am asking that can I solve this question using coulomb's law and some other textbook stuff like properties of conductors, etc.?

I mean, if I don't know the concept of electric field (which I am still trying to justify, even if for a sphere), can I solve it? If so, then what should I do? (Coulomb's Law?)

4. Jul 19, 2012

### TSny

A direct integration using Coulomb's law is a bit messy. Using the following concepts involving electric field makes it a lot easier:

(1) The relation between force and electric field: F = qE

(2) The relation between electric field at the surface of a conductor and the charge density: E = σ/εo

(3) The electric field produced by a large flat sheet of uniform charge density: E = σ/2εo

The latter 2 properties are easily derived from Gauss' law.