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My understanding of electrostatics is based on the concept of charge and electric field. If each point in a region of space has a certain charge density associated to it, then by integrating Coulomb's law over space, we get a certain electric field vector at each point.

For example, I would describe a typical "two parallel plates" problem as two parallel plates, each of which has a charge density of 1 Coulomb per square inch.

However, a way of describing electrostatics problems that I see sometimes involves talking about voltages. Now, I understand voltage as the potential of an electric field, but I don't understand what it means to say that we have two parallel plates, and that one is "charged to 5V" and the other is "charged to 0V".

In terms of charge density functions and the electric fields that they generate, what does it mean to declare that a pair of plates is "charged to a potential difference of x volts"?

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    $\begingroup$ If you know the capacitance of two parallel plates, then you can interconvert between charge and voltage. $\endgroup$ – probably_someone May 4 '18 at 9:08
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The voltage between the plates, the charges on them and the separation and medium in between are related by the following formula $$Q = C U$$ where $Q$ is the total charge on the plate, $U$ the voltage difference and $C$ the capacitance (it depends on the plate separation and the dielectric between the plates).

From these quantities it is immediately obvious that a certain voltage between the plates corresponds to a certain number of charges or charge density on the plates (all other things being equal).

Adding more charges to the plates for example will result in a higher voltage difference, which you can also intuitively understand from the fact that the electric field between the plates becomes larger according to $$U = Ed = \frac{Qd}{\varepsilon A}$$ where $d$ is the plate separation, $E$ the electric field, $\varepsilon$ the dielectric permittivity between the plates and $A$ the area of the plates.

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The terms voltage and potential are often used interchangeably and the exact meaning is inferred from the context.

The potential is defined relative to infinity and therefore, in a sense, could be viewed as an absolute value, whereas the voltage is defined as a difference of potentials between two points and therefore could be viewed as a relative value.

When we say that a plate is charged to 5V, it should be clear from the context what it is relative to or what the reference point is, which could be another plate, the ground or infinity.

When we declare that a pair of plates is "charged to a potential difference of x volts", it means that the voltage on one of the plates relative to the other will be xV or -xV depending which of the plates is chosen as a reference point.

If we measured the voltage on each of the plates relative to any point in space, the difference between those two voltages would be xV as well.

The charge densities on these plates needed to produce xV could vary. For instance, we can charge one plate and not charge the other plate at all. Or we can charge both plates with the charge of the same sign but different magnitude. Or we can charge the plates with charges of different signs.

In all those cases, the field between the plates will be roughly the same in a sense that it would add or subtract x eV to or from the energy of an electron moving from one plate to another.

A plate capacitor represents a special case, when electrons from one plate are moved to the other and therefore the plates end up with equal and opposite charges.

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