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Does there exist a single plate capacitor(conductor)? if yes

How will you define the capacitance and potential(difference) of such conductor?

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Wouldn't static electricity (for example on a balloon) count as a single plate "capacitor"? – fibonatic Feb 11 '14 at 7:26
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The term you're looking for is self-capacitance. Look it up, you'll get some insight. – Nanite Feb 11 '14 at 7:32

A simple example is that of a sphere. One way to find its capacitance is to take the limit of a nested sphere capacitor with radii $a,b$: $$C = \lim_{b\to\infty}\frac{4\pi\epsilon_0}{\frac{1}{a}-\frac{1}{b}} = 4\pi a\epsilon_0\text{.}$$ A van de Graaff generator is a commonly discussed in physics classes, and involves this type of setup.

For a parallel-plate capacitor, however, doing the same gives zero capacitance.

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I think you get zero when calculating the limit of the parallel plate because the parallel plate capacitance formula of $\varepsilon A/d$ is an approximation that neglects the very small self capacitance of the individual plates and only considers the mutual capacitance. That's why it is typically said to hold only when $A>>d^2$. With one plate at infinity there is no mutual capacitance and having simplified the self capacitance away there is nothing left. This simplification isn't made with concentric spheres because it is easier to calculate the fields in spherical coordinates. – Omegaman Jan 31 at 7:03

A single conductor also possess capacity to store charge. It may be treated as parallel plate capacitor, whose one plate is at infinity.

If this doesn't help, comment on the part where you have problem.

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Yes. Capacitance is very well defined for a sphere in vacuum, and can be extended to other media and shapes. The following is a simple abd correct definition of capacitance. Adding other conductors disturb the capacitance of a single conducting body, giving rise to mutual capacitance.

The following is exerpted and edited from The Great Soviet Encyclopedia (1979). Search that term for more complete answer.

Capacitance

The capacitance C of an isolated conductor is equal to the ratio of the charge on the conductor to the conductor’s potential.

Capacitance is determined by the size and shape of a conductor and by the electrical properties of the surrounding medium, that is, by the medium’s dielectric constant. Capacitance of a conducting sphere in vacuo is equal to the radius of the sphere. Capacitance is measured in centimeters in the cgs system and in farads (1 farad = 9 × 10^11 cm) in the International System of Units (SI).

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