Self-Capacitance of an Insulating vs. Conductive Sphere

[Apogee]

Hey, guys! I had a brief electromag question.

If capacitance is defined as the amount of charge that can be stored in a capacitor per unit of potential difference, then technically can capacitors made from insulators still have a capacitance? The problem I'm considering is calculating the self-capacitance of a sphere in a vacuum. I first solve for the potential difference and then use the definition of capacitance to cancel the charge term and just have capacitance.

However, if one considers the sphere to be conducting, then the charge is only stored on its surface. However, if the sphere is considered to be insulating, then the same charge is distributed throughout the sphere. If the charge density is assumed to be uniform, does it make sense to consider the problem with an insulating instead of conductive sphere?

 

[just dani ok]

then you would have an equivalent of a capacitor connected in series with an extremely high resistor, which is not very useful.

 

[Apogee]

I'm not sure how you came to that conclusion. Care to elaborate?

 

[NascentOxygen]

Hey, guys! I had a brief electromag question.

If capacitance is defined as the amount of charge that can be stored in a capacitor per unit of potential difference, then technically can capacitors made from insulators still have a capacitance? The problem I'm considering is calculating the self-capacitance of a sphere in a vacuum. I first solve for the potential difference and then use the definition of capacitance to cancel the charge term and just have capacitance.

However, if one considers the sphere to be conducting, then the charge is only stored on its surface. However, if the sphere is considered to be insulating, then the same charge is distributed throughout the sphere. If the charge density is assumed to be uniform, does it make sense to consider the problem with an insulating instead of conductive sphere?

EDIT: The field outside the conductive sphere is comparatively easy to calculate, it is indistinguishable from the field you'd get by having the total surface charge concentrated as a point source located at the centre of that spherical region. Contrast this with the distributed charge model with charge embedded throughout the dielectric making it of dubious practicality, and being not nearly so easy to evaluate.

So in essence I'd say there'd be no difference.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

 

[just dani ok]

I'm not sure how you came to that conclusion. Care to elaborate?

it was just a hindsight based on following reasoning. a conductor as well as isolator can store electric charge. eg. rubber balloon, plastic ruler, metal leaf electroscope. hence you will get a capacitor.
but if your material is isolator then it would be difficult for the charge to move from one part to the other, which is analog to high resistor.
you can assume that the carges could be distributed evenly inside the isolator sphere, the problem is, how can you move them to inner part of the sphere in the first place, since it is an isolator?

 

[Apogee]

Gauss's Law says that the field outside any spherical region is determined solely by the sum of the charges enclosed and not by how they are distributed, and that this field is indistinguishable from the field you'd get by having the nett enclosed charge concentrated as a point source located at the centre of that spherical region.

So in essence I'd say there'd be no difference.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

That's true. But if you take the potential at the center of the sphere, it changes because you now have a field inside the sphere. Inside the sphere at a certain radial distance from the center, you enclose charge, and, by Gauss's Law, you will have a field. Therefore, you will still gain potential energy for every charge that you bring to the center of the sphere because you have to continue to do work against an electric field. The potential will be larger in this case.

Does it make sense to consider the potential at the center of these spheres rather than the surface? In the case of the conductor, since there is no field inside, the potential will be the same whether we consider the potential at the surface or the center. In the case of the insulator, it was established that it will be slightly different (I think larger by a factor of 3/2). So, in this case, if it makes sense to consider the self capacitance with the potential taken at the center rather than the surface, the self-capacitances of each would be different.

 

[NascentOxygen]

If you go back to fundamentals, you are finding the potential of the object by considering the sum of all those energies involved in moving each individual charge from infinity along its individual flux line to its particular location about the sphere. In the case of a conductive shell, symmetry and Gauss's Law simplifies your calculations considerably. But where the charges are embedded throughout the volume of the sphere, if there is no simplification you can find yet still needing to sum the individual energies of all the charges involved, then you may have no alternative but to use integral calculus in 3 dimensions to do that summation. But I'm afraid this takes me into realms I've long forgotten, so I'll have to leave it for others to help you beyond this.

 

[nasu]

Gauss's Law says that the field outside any spherical region is determined solely by the sum of the charges enclosed and not by how they are distributed, and that this field is indistinguishable from the field you'd get by having the nett enclosed charge concentrated as a point source located at the centre of that spherical region.

So in essence I'd say there'd be no difference.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

Gauss' law does not say this. It says that the (total) flux through the surface is independent of the distribution of charges.
A given flux can be produced by any number of electric field configurations so the field do depend on the charge distribution.
You may be thinking about the field of an uniformly charged sphere which is indeed the same as the field of point charge located in the center.
As the words in bold show, this is a statement valid only for a very specific configuration of charges.
If what you said would be true in general, the point charge did not have to be in the center, did it?

@Apogee
The problem is not if you can but if it would be useful a capacitance defined in such a way.
For a conductive object you can attach always a unique potential. This is not true for an insulated sphere.
In general all points of the objects may have different potentials. Which one will you use to define "capacitance"? For a uniformly charged sphere you may take the potential of the surface.
But then the second problem is that if you take an uncharged sphere and attach it to a source of charge, at potential V, it will not charge with the Q=CV, uniformly distributed.

 

[NascentOxygen]

Gauss' law does not say this. It says that the (total) flux through the surface is independent of the distribution of charges.

Indeed, I was thinking of flux.

The problem is not if you can but if it would be useful a capacitance defined in such a way.
For a conductive object you can attach always a unique potential. This is not true for an insulated sphere.

The device would have limited usefulness, offering permanent static storage but useless for dynamic applications. Nevertheless, if it is viewed as a collection of minute capacitors, and we know the work done in creating that distribution of charge and the total charge, we can arrive at an effective value for C. Work input would allow a direct comparison with the work involved in charging a conductive sphere for the same charge storage.