Investigating the Charge Distribution on a Bulging Sphere

[Daniel777]

Homework Statement

Due to nonuniform mechanical strength, when a conducting sphere is
given a charge Q , the electrostatic forces make a small hemispherical
bulge on its surface. The radius r of the bulge is negligibly small as
compared to the radius R of the conducting sphere.
Can you assume charge on the bulge distributed uniformly?

Relevant Equations

Charge distribution

In this question it is given that the sphere which is conducting is initially given a charge q then due to nonuniform mechanical strength and due to electrostatic force it creates a Small hemispherical bulge on its surface?

okay my doubt is Let me define a term σ where σ is surface density density so is dq/da going to be constant? And also on bulge part how would the charge distribution be is it going to be uniformly distributed over bulge part? According to me initially when sphere was given a charge q it was uniformly distributed but due to electric forces charges repel each other and create a bulge and I think they should repel each other and settle down in such a way that then also charge will be uniformly distributed only? Is it that?

I am really confused I can't be able to interpret the results? Please help me out?

 

[berkeman]

Hopefully you will get better replies from others, but it sure seems like if the growth in radius of the bulge is negligible compared to the sphere's starting radius, that the divergence would not deviate much in that hemisphere compared to the slightly smaller hemisphere. Maybe I'm missing something, though...

https://mypages.iit.edu/~smile/guests/gsmxsec1.htm

 

[haruspex]

okay my doubt is Let me define a term σ where σ is surface density density so is dq/da going to be constant? And also on bulge part how would the charge distribution be is it going to be uniformly distributed over bulge part? According to me initially when sphere was given a charge q it was uniformly distributed but due to electric forces charges repel each other and create a bulge and I think they should repel each other and settle down in such a way that then also charge will be uniformly distributed only? Is it that?

I am really confused I can't be able to interpret the results? Please help me out?

I'm not sure whether you are saying the density should still be roughly the same over the whole sphere or just over the bulge (and different from that over the rest of the sphere).
Crudely, let's suppose the second case, ignoring variations over the bulge and local changes near the bulge.
For a given charge density, how does the surface potential vary with the radius of the sphere? For the whole surface of the sphere to be still at one potential, what does that suggest about the density on the bulge?

 

[Steve4Physics]

it sure seems like if the growth in radius of the bulge is negligible compared to the sphere's starting radius, that the divergence would not deviate much in that hemisphere compared to the slightly smaller hemisphere.

Since r<<R, we can consider a ‘close-up’ view to get an intuition. The bump' and surrounding region is then, in effect, a conducting hemisphere attached to a flat conducting surface.

I interpret the question to be “Can you assume the charge on the hemisphere is distributed uniformly over the hemisphere?“.

 

[haruspex]

I interpret the question to be “Can you assume the charge on the hemisphere is distributed uniformly over the hemisphere?“.

I was thinking more that it clearly cannot be (and likewise the distribution on the nearby plane/large sphere), but can it be so treated as a first approximation in order to get the relationship between the two densities ?

 

[Steve4Physics]

I was thinking more that it clearly cannot be (and likewise the distribution on the nearby plane/large sphere),

I agree.

but can it be so treated as a first approximation in order to get the relationship between the two densities ?

Possibly. interesting. I'm not sure though.

But the original question is:
"Can you assume charge on the bulge [is] distributed uniformly?"
and that seems a fairly specific/unambiguous question.

 

[haruspex]

But the original question is:
"Can you assume charge on the bulge [is] distributed uniformly?"
and that seems a fairly specific/unambiguous question.

I feel that can be read as can we take it to be uniform for some (unstated) purpose?

 

[Steve4Physics]

I feel that can be read as can we take it to be uniform for some (unstated) purpose?

Yes, that's a possible interpretation. I wonder if @Daniel777 (the OP) can confirm the question posted is accurate and complete. And if the official answer becomes available, it would be interesting to see it.

 

[Daniel777]

over the bulge and local changes near the

Yes, that's a possible interpretation. I wonder if @Daniel777 (the OP) can confirm the question posted is accurate and complete. And if the official answer becomes available, it would be interesting to see it.

This is actually the original question.I hope it doesn't have any error.

 

[berkeman]

Oh, it's a little bump, not the whole hemisphere of the overall sphere. Sorry that I misunderstood the question.

 

[Daniel777]

Oh, it's a little bump, not the whole hemisphere of the overall sphere. Sorry that I misunderstood the question.

Yeah but I am still in determining how would the charge configuration be over the entire sphere including the bulge part and electric field lines direction

 

[berkeman]

Is the divergence of D higher or lower above the bump compared to the rest of the sphere?

 

[Daniel777]

Is the divergence of D higher or lower above the bump compared to the rest of the sphere?

I haven't studied about the divergence yet

 

[haruspex]

Yeah but I am still in determining how would the charge configuration be over the entire sphere including the bulge part and electric field lines direction

Typically charge distribution is a complicated matter except in some special cases. I feel confident saying It would not be uniform on either curved surface, but to solve the next part I think you have to assume it is near enough uniform. At least try that.

 

[PhDeezNutz]

https://indjst.org/articles/capacitance-of-two-overlapping-conducting-spheres

Here’s a free paper on applying the method of images to overlapping spheres.

 

[haruspex]

https://indjst.org/articles/capacitance-of-two-overlapping-conducting-spheres

Here’s a free paper on applying the method of images to overlapping spheres.

I couldn’t see how the method of images could be applied to hemispheres, though. Do you see a way?
Besides, I am only aware of such methods being used to find induced potentials. Finding a charge distribution is another matter.

 

[PhDeezNutz]

I couldn’t see how the method of images could be applied to hemispheres, though. Do you see a way?
Besides, Inam only aware of such methods being used to find induced potentials. Finding a charge distribution is another matter.

I may have just instinctively and prematurely drawn a comparison where there is none. I’m going to work on it…..maybe I’ll come up with something useful or maybe it will turn out that I was incredibly misguided.

Anyway thanks for challenging it because now I have an interesting problem to work on.

 

[Steve4Physics]

Since @Daniel777 hasn’t met divergence yet, it sound like they are [choosing political correctness over grammar] working at an introductory level. That suggests only a relatively simple approach is being called for.

@Daniel777, have you met the effect of curvature on surface charge density (of conductors)? For two conducting spheres at the same potential, there is a simple relationship between the spheres’ surface charge densities and their radii.

(The classic example is that a sharp point on a charged conductor has a high surface charge density compared to the rest of the conductor’s surface.)

If you assume the average surface charge densities on the hemisphere and on the sphere obey this relationship, you might be able to answer part (b) of the question.

 

[haruspex]

Since @Daniel777 hasn’t met divergence yet, it sound like they are [choosing political correctness over grammar] working at an introductory level. That suggests only a relatively simple approach is being called for.

@Daniel777, have you met the effect of curvature on surface charge density (of conductors)? For two conducting spheres at the same potential, there is a simple relationship between the spheres’ surface charge densities and their radii.

(The classic example is that a sharp point on a charged conductor has a high surface charge density compared to the rest of the conductor’s surface.)

If you assume the average surface charge densities on the hemisphere and on the sphere obey this relationship, you might be able to answer part (b) of the question.

This is what I have been trying to get @Daniel777 to try in posts #3 and #14.