Two Conducting Spheres connected by a wire

[tomizzo]

Homework Statement

Two conducting spheres of radii rA and rB are connected by a very long conductive wire. The charge on sphere A is Qa and rA < rB.

What is the charge on sphere B?

Which sphere has the greater electric field strength immediately above its surface.

Homework Equations

Esurface = $\eta$/$\epsilon$naught

The Attempt at a Solution

So I assume that any charge placed on the two conductors will reach an equilibrium which I assume would mean that both conductors have the same surface charge density.

That is:

Qa/(4$\pi$*rA^2)=Qb/(4$\pi$*rB^2).

However, when I solve for Qb, I get (rB/rA)^2*Qa which is incorrect. The correct answer is apparently (rB/rA)*Qa.

For the second question, I need to find the electric field strength at the surface of each conducting sphere. Well since I assumed that the surface charge density was the same, using the equation listed above, I should have equivalent electric field strengths. Instead, the answer states that the electric field strength on sphere A is greater than that of B. Why?

 

[SammyS]

Homework Statement

Two conducting spheres of radii rA and rB are connected by a very long conductive wire. The charge on sphere A is Qa and rA < rB.

What is the charge on sphere B?

Which sphere has the greater electric field strength immediately above its surface.

Homework Equations

Esurface = $\eta$/$\epsilon$naught

The Attempt at a Solution

So I assume that any charge placed on the two conductors will reach an equilibrium which I assume would mean that both conductors have the same surface charge density.

That is:

Qa/(4$\pi$*rA^2)=Qb/(4$\pi$*rB^2).

However, when I solve for Qb, I get (rB/rA)^2*Qa which is incorrect. The correct answer is apparently (rB/rA)*Qa.

For the second question, I need to find the electric field strength at the surface of each conducting sphere. Well since I assumed that the surface charge density was the same, using the equation listed above, I should have equivalent electric field strengths. Instead, the answer states that the electric field strength on sphere A is greater than that of B. Why?

There is no valid reason to conclude that the surface charge densities are equal.

The spheres are connected by a wire (conductor). What does that imply about the electric potential of each?

 

[tomizzo]

There is no valid reason to conclude that the surface charge densities are equal.

The spheres are connected by a wire (conductor). What does that imply about the electric potential of each?

That would mean that the electric potential will be the same for each sphere.

So I'm starting to believe that the surface charge densities are not equal. However, why is this? If I were to put charge on a conductive plate, I would think that the charge repel each other and since it's a conductor, the charge would spread out equally across the plate. Thus giving a single surface charge density.

Is this assumption incorrect?

 

[SammyS]

That would mean that the electric potential will be the same for each sphere.

Yes. That's correct.

So I'm starting to believe that the surface charge densities are not equal. However, why is this? If I were to put charge on a conductive plate, I would think that the charge repel each other and since it's a conductor, the charge would spread out equally across the plate. Thus giving a single surface charge density.

Is this assumption incorrect?

Yes. The assumption is incorrect.

 

[tomizzo]

Yes. That's correct.


Yes. The assumption is incorrect.

I've tried to be as logical as I could about this. Do you care to elaborate on how this is incorrect?

 

[SammyS]

I've tried to be as logical as I could about this. Do you care to elaborate on how this is incorrect?

What do we know about the charge distribution on an irregularly shaped conductor and the electric field near its surface?


Excess charge tends to accumulate in regions with smallest radius of curvature (in a convex sense) with less density where the radius of curvature is larger and even less dense in locally flat or concave regions.

The spheres are far apart. I expect that's so that the electric field from one doesn't much affect the other.

 

[Chestermiller]

If Q_A is the charge on sphere A, what is the electrical potential at the surface of sphere A (relative to infinity)? If Q_B is the charge on sphere B, what is the electrical potential at the surface of sphere B (relative to infinity)?

Chet