# B The capacitance of an isolated sphere

#### pager48

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#### hutchphd

The capacitance of a sphere really means the capacitance of the isolated sphere and the rest of the world (usually taken as a conducting sphere at infinity.....its a perfectly good approximation) when charge is separated. So what is the definition of capacitance???

#### Henryk

Gold Member
isolated spheres have capacitance but how can it be measured with an instr
Very good point. An isolated sphere has a well defined capacitance but to measure anything one would have to connect an instrument and that means the sphere would no longer be isolated.

#### lesaid

How about taking two identical conducting spheres, well separated. Connect each one individually to a known voltage relative to an 'absolute ground'. Not an arbitrary reference, but a ground where the numbers of positive and negative charges are equal.

Each sphere will take on a charge corresponding to that voltage and its capacitance.

Now disconnect the spheres from the charging circuit, keeping them electrically isolated from anything else, bring them close together and measure the repulsive force between them. From this, the charge of each sphere can be calculated. The original 'charge' voltage is known, so the capacitance can be calculated with no electrical connection or additional nearby conductor required.

Alternatively, measure the charge transferred while 'charging' the spheres.

A calculation equating the electrostatic potential required to reach equilibrium at a given voltage between a test charge in the connecting wire and a test charge on the sphere's surface so that no current flows leads to the corresponding charge on the sphere, and from there, a derivation of the standard formula for the capacitance of an isolated sphere.

#### pager48

Connecting any instrument makes the sphere have no capacitance?

#### lesaid

The sphere would still have capacitance, but so would whatever was connected to it, both between the connected 'thing' and the sphere, and between the connected 'thing' and the surrounding environment. So the measured capacitance is influenced by the measuring device.

In my suggestion, when charging the spheres (while they are well isolated), there will be a very tiny capacitance from the (thin) wire that connects the sphere to the charging voltage, and that must have some small effect. But then so does the fact that this whole experiment is being carried out in a room on earth rather than in the middle of an otherwise empty universe. All this is only an approximation to a truly isolated sphere, though I think, it can be a very close approximation.

The idea behind my suggestion is that we are not connecting anything to measure the capacitance. We are instead measuring the effect of the charges on the two spheres directly, without requiring an electrical connection, Also, the capacitance of the spheres will be affected by their proximity but the stored charge won't be, as the spheres are then truly electrically isolated. The voltage on the surface of the spheres will change because of that change in capacitance, but the electrostatic repulsion is dependent on the charges present, not the associated voltages. So as far as I can see, the measurement should be unaffected.

Though I imagine one might have to take into account the change in charge distribution on the spheres. As the spheres approach each other, the charges would no longer be centred on the geometrical centre of the spheres, but would migrate towards the areas of the spheres farthest away from each other. The measurement of electrostatic force should be made while the distance between the spheres is large compared to the size of the spheres, but I think it should be possible to account for and eliminate this effect from the calculations.

#### pager48

According to the isolated sphere capacitance formula better dielectric means more capacitance.

Were is the dielectric located in an isolated sphere capacitor?

In my suggestion, when charging the spheres (while they are well isolated), there will be a very tiny capacitance from the (thin) wire that connects the sphere to the charging voltage, and that must have some small effect.
But any instrument adds some sort of error why is measuring isolated sphere capacitance any different than measuring ordinary capacitors?

#### lesaid

An 'ordinary' capacitor generally consists of two electrodes very close together separated by a dialectric. The actual capacitance is dominated by the geometry of those components and the material of the dialectric. Whatever is around that capacitor has little effect in comparison.

However, an isolated sphere can be thought of as one electrode in a very low value capacitor, the other electrode of which is the environment around it. Changes in that environment, including the proximity of any other conductor, can have a very significant effect. Even the proximity of an insulator can also have a real impact, if it has a different dialectric from air.

This situation is exploited in capacitive sensors such as many 'touch' sensors.

So more care needs to be taken to measure the capacitance of an isolated sphere than with an ordinary capacitor.

#### pager48

Is the isolated sphere capacitors dielectric inside the sphere?

#### lesaid

No - it is the air around the sphere, the moisture in that air, anything other insulator near the sphere will contribute to the dialectric.

I believe that, for example, humid air around the sphere would result in a slightly higher capacitance than dry air, Though I stand to be corrected in that if any experts care to chip in.

#### pager48

No - it is the air around the sphere, the moisture in that air, anything other insulator near the sphere will contribute to the dialectric.

I believe that, for example, humid air around the sphere would result in a slightly higher capacitance than dry air, Though I stand to be corrected in that if any experts care to chip in.
Then it does not matter if the sphere is hollow or solid to be a capacitor?

#### hutchphd

Then it does not matter if the sphere is hollow or solid to be a capacitor?
So long as the charge is uniformly distributed on its surface (usually in practice because it is a conductor).

#### lesaid

It may be interesting to know that the earth itself as an isolated sphere has a capacitance of about 710 uF.
Whereas the sun is around 77,000 uF and a 10 cm sphere would be around 11 pF.

#### pager48

Isn't for a 10cm sphere its

4pi*50 = ~628pF ?

#### lesaid

You have the correct formula for the capacitance of an isolated sphere - i.e. $C=4\pi\varepsilon_0 R$. Plug in the following values

$\varepsilon_0=8.85\times 10^{-12} F m^{-1}$ (the 'permittivity of free space')
$R=0.1 m$

This comes out to $1.11\times 10^{-11} Farads$ or $11.1\times 10^{-12} F$ which is to say, $11.1 pF$

#### hutchphd

It may be interesting to know that the earth itself as an isolated sphere has a capacitance of about 710 uF.
It is my understanding that the earth surface and the ionosphere form a concentric spherical capacitor that is largely charged by thunderstorm processes and this gives rise a ~uniform static field of several hundred volts/meter on average. The capacitance of that system should be roughly your number times 100 (i.e. REarth/ionosphere altitude). Also the outer surface is actually the ionosphere !

#### pager48

You have the correct formula for the capacitance of an isolated sphere - i.e. $C=4\pi\varepsilon_0 R$. Plug in the following values

$\varepsilon_0=8.85\times 10^{-12} F m^{-1}$ (the 'permittivity of free space')
$R=0.1 m$

This comes out to $1.11\times 10^{-11} Farads$ or $11.1\times 10^{-12} F$ which is to say, $11.1 pF$
Isn't permittivity in air 1 making capacitance ~628pF?

#### lesaid

I think you might be mixing up "relative permittivity' with 'absolute permittivity'. The relative permittivity of vacuum, and to a close approximation, air, is indeed (by definition) '1'. However, the capacitance formula calls for the absolute permittivity.

Take a look at https://en.wikipedia.org/wiki/Permittivity which explains this more fully.

#### pager48

In that case the only ways to increase capacitance is bigger sphere or coating sphere with high K value dielectric material?

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#### lesaid

Yes, or have a grounded conductor nearby, so the sphere is no longer isolated.

If you want to make a capacitor, an isolated sphere is perhaps not the most effective way!

I don't know how effective coating the sphere would be. I am sure it would raise the capacitance, but I haven't tried to figure out by how much. My guess is, not by a huge amount. This size of the sphere is probably much more significant.

But you've got me curious about that now :)

#### pager48

Would spraying the sphere with adhesive spray and dipping it into barium titanate or calcium copper titanate powder making several coats raise the spheres capacitance? There is also titanium dioxide powder which is much less expensive.

Can you describe the nearby grounded conductor arrangement and how it increases capacitance?

#### lesaid

Off hand, I really don't know how much such a coating would influence the capacitance. But I would guess, not by a lot. It's something I'm now interested in figuring out, though I haven't got time at the moment!

Can you share with us what you are trying to achieve here? I have the impression you are wanting to design or build something?

If the idea is associated with high voltages, let me warn you that significant capacitance can be extremely (lethally) dangerous! That's one reason why Van de Graaf generators with large spheres are not a good idea unless you really know what you are doing. (the other reason is that larger spheres have less curvature which allows higher voltages to accumulate without so much loss - higher voltages in conjunction with higher capacitance are even more dangerous!)

If you want a capacitor though, in general, an isolated sphere isn't a good way to make one!

#### pager48

Can you share with us what you are trying to achieve here? I have the impression you are wanting to design or build something?
Going to investigate figure 4 from attached publication.

But I would guess, not by a lot. It's something I'm now interested in figuring out, though I haven't got time at the moment!
Maybe there is a publication baring that information but its likely buried somewere and its emphisis is likely not about isolated spherical capacitors.

The attached publication has an attached conductor which is grounded you mentioned attached grounded conductors can increased capacitance. Is the attached conductor similar to what you were describing?

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#### lesaid

Thanks for posting that article. I shall enjoy reading it!

And great that you are keen to investigate physics at home. I too enjoy home experimentation in 'practical' physics (have a thread about a Foucault pendulum in the 'projects' area of this forum). It is a great way to learn!

Looking at 'figure four', describing a system for electrical transmission through a single wire, consider the spherical 'capacity' symbol at the right side of the diagram, and think what that really represents. In the hydraulic analog, the sphere equivalent is labelled as 'elastic reservoir'. Think of this as a balloon. As the piston goes back and forth, the balloon inflates and deflates, air passing through the 'friction device' back and forth, from which energy is extracted to 'light the bulb'.

So far, so good. But there is a critical piece missing from Tesla's analogy. Take a look at the first diagram in the attached file. When the piston moves to the right, the balloon expands. But what is not mentioned is that it displaces air which replaces that sucked in behind the moving piston. There is a corresponding (small) flow of air through the atmosphere from the balloon to the piston. This is effectively a hidden return path.

Now look at the electrical diagram. The sphere is acting as a capacitor, storing and supplying energy alternately as it charges and discharges. But a capacitor has two ends and this sphere is no exception. The other end is the ‘ground’. Take a look at the second diagram in the attached file, where I have drawn in a capacitor to represent the situation more correctly. The ground itself is conductive and acts as a resistor. This is a return path, equivalent to the atmosphere in the hydraulic analogy.

Now to answer your question about having a grounded conductor near the sphere. This is equivalent to mounting the sphere close to the ground. The effect is to increase the capacitance to ground. You could get a better result by forgetting about the sphere and using a regular capacitor to ground, of a higher capacitance.

Or you might prefer simply to forget the capacitor and simply ground the sphere itself – using the direct resistive path through ground as the return without the capacitor at all.

Here is an example of this in practice

https://en.wikipedia.org/wiki/Single-wire_earth_return

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