What Force Is Needed to Hold a Cut Charged Sphere Together?

[Dinheiro]

Homework Statement

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

Homework Equations

Elestrostatic equations

The Attempt at a Solution

After the cut, the charges would get redistributed along the section, right? I figured a spherical cap repelling the other piece as they would get the same charge signal, so the necessary force to hold them both is the repelling force F,
F = EQ
E is the electric field
How can I calculate this E through the spherical cap??

 

[rude man]

I wouldn't think in terms of an actual cut. The two quasi-hemispherical sections are joined together so all the charge is on the joined sphere's round surface, but there is a repulsive force trying to separate the two sections.
Beyond that I have no hints to offer.

 

[Dinheiro]

Good call, as the force gets the quasi-hemisferical sections together, the charges will remain in the sphere's surface. I will retry it later.

 

[rude man]

Good luck with this, I suspect it's a tough problem, probably belongs in the advanced physics forum.

WAG: there's a line thru the center of the sphere and perpendicular to the cut. For each sphere section there's a point on this line where the net force in the direction of the line is zero (taking one section at a time). If you were to put the respective surface charges Q1 and Q2 at those two points, could you argue F = k Q1 Q2/d^2 where d is the distance separating those two points?

 

[HalfLostAndSearching]

I would approach this problem by first reviewing the equations for electrostatics, particularly those related to electric fields and forces. I would also consider the principle of superposition, which states that the total electric field at a point is the sum of the electric fields from all individual charges.

To calculate the electric field at a point due to a charged spherical cap, I would use the equation for electric field of a point charge, E = kQ/r^2, where k is the Coulomb's constant, Q is the charge on the spherical cap, and r is the distance from the center of the cap to the point where the electric field is being measured.

Next, I would consider the distribution of charges on the two parts of the sphere after it has been cut along the plane. Since the sphere was uniformly charged before the cut, the charges on each section would also be uniformly distributed. This means that the electric field at any point on the surface of the sphere would be the same, regardless of which section it belongs to.

To calculate the force necessary to hold the two parts of the sphere together, I would use the principle of superposition to sum up the electric fields from both sections at the point where they are closest to each other (at a distance of h from the center). This would give me the total electric field at that point, which I can then use to calculate the necessary force using the equation F = EQ.

Overall, to accurately calculate the force necessary to hold the two parts of the sphere together, I would need to consider the distribution of charges on both sections, the distance between them, and the electric fields from each section at the point of closest approach.