Capacitance only geometry dependent?

[nassboy]

Is there a proof that states that if a system only contains perfect conductors, dielectrics and voltage sources that the capacitance between any two conductors is only geometry dependent?

 

[atyy]

A capacitance that depends only on geometry is definable only in electrostatics. In electrostatics the relation between charge and potential is given by Poisson's equation, whose solution depends on boundary conditions (ie. zero potential on the geometric surfaces of the conductor) http://hyperphysics.phy-astr.gsu.edu/hbase/electric/laplace.html. You can see that if you multiply the charge by a constant, then the potential will be multiplied by the same constant to solve Poisson's equation, so capacitance is dependent only on the geometry of the conductors.

 

[nassboy]

I kind of see what you are saying from the differential equations, but it is hard to see it qualitatively for me. If I look at a system with 3 conductors placed at 3 different potentials, I can't visualize how the capacitance between two of the three is not changed by the potential of the third...but is changed by the 3rd conductor being there.

 

[atyy]

In full electrostatics, the capacitance of the whole system is changed by the 3rd conductor - it's the geometry of the whole system.

In circuits, one usually assumes that the capacitors are far away enough from each other that they don't affect each others capacitance.

 

[atyy]

BTW, if you have ever tuned a radio to a particular station, only to find it mistuned once you remove you hand from the dial, that's because you are the third conductor - and the tuning of the radio depends on the exact position of you.

 

[Kibuzo]

I kind of see what you are saying from the differential equations, but it is hard to see it qualitatively for me. If I look at a system with 3 conductors placed at 3 different potentials, I can't visualize how the capacitance between two of the three is not changed by the potential of the third...but is changed by the 3rd conductor being there.

When i studied it i tried to imagine the capacitance as the capability of a system to "hold fields", because you always want to stack charges, but only a strong field can give you the possibility to do that.
Now you have that the field depends on the surface and on the distance between plates and the distance between them depends on their position and their geometry (the distance of every point of the surface).
You would never want to waste any line of field so try to get every line ending on a plate, so that you wan't have any loss, and this is why, even if they don't really exist, spherical capacitors looks that nice, every line of field that leaves the inner sphere always lands on the outer sphere. Real capacitors are most commonly made of two long parallel plates rolled in a cilyndrical shape, but you probably already know this, and anyway it is not the thing you are asking for.

In italian we use another word for the capacitor, that instead of being related to "capacity" it is releted to "density", something like "densifier": i always imagined it like something that increased the density of field, and always though of the capacitor when i had to think of it as a system holding a lot of charges, like in a tank holding a lot of fluid.

At this point it should be clear that it depends on the geometry of the materials, on their distance etc. Two observation.
1) if i have a dielectric the capacitance increases. This, following what i said, is due to the fact that you have the field lowered by the dielectric, but using the superposition property of the electric field there is a "lowering" field stored on the dielectric (a lot of - charges put themeselves close to the + plate, and vice versa. At this point you have something like this: +- +- so that you can consider 2 different fields superpositioned, more fields more capacity).

2) I'm still a student and i worked this out completely by myself, so something i said might be wrong, don't believe mee too faithfully. But this model always worked for me, it is nothing "official" but as it works most of times i always believed it to be good, and i keep using it every day with good results, so it might be your thing, just in case you want to have an instant shot of a capacitor every time you see one.

Hope that helps.

 

[nassboy]

Is it possible to set up a more formal proof of this? I'm still struggling a little.

 

[atyy]

Sorry, the multiple conductor situation may not be true, unless there's an additional condition on the fraction of total charge on each conductor - not sure what the exact statement should be.

 

[nassboy]

The statement seems to be true for any number of conductors. I've been playing around with different configurations in an electrostatics simulator.

 

[atyy]

Try looking up coefficients of potential here: http://books.google.com/books?id=4liwlxqt9NIC&printsec=frontcover