Capacitance

These notes focus on the physics behind capacitance. For use of a capacitor, see capacitor.

Capacitance

Suppose the total charge on a conductor is and electric potential is . We add some extra charge so it increases by a factor of .
The extra charge will increase the surface charge density everywhere by the same amount without changing the charge distribution.

Thus, the E-field and electric potential should increase by the same amount . We can conclude that the potential on a conductor is proportional to the charge on the conductor.

where is the proportionality constant between the charge and electric potential.

Equation

This defines capacitance for us:

Definition

Capacitance is the electric charge that must be added to the conducting body to increase its electric potential by 1 volt.

Capacitance is measured in farads.

Example

A spherical conductor of radius has charge deposited on it. What is the capacitance of this system?

solution
We know the potential at the surface is:

Thus,

where is Coulomb's constant.

Capacitors

Suppose charge gets transferred to the conductor connected to the positive terminal of the source. Then will be transferred to the negative terminal.

Basic Configuration of a Capacitor

Thus a potential difference is created across the two conductors, . The potential difference is related to the magnitude of on the conductor.

Equation

Remark

The capacitance is only dependent on the geometry of the conductors, and the permittivity of the Dielectrics between them. It does not depend on the charge . Mathematically, we expect that should cancel out somewhere along the line.

In addition, a capacitor will have capacitance even without voltage or charge, similar to how objects in space still have mass without gravity.

A capacitor does not have to be only 2 Parallel Plates. We can have multiple plates to increase capacitance, or even one place (such as a capacitive touch screen).

Important

We connected a DC source to the system which deposited charges onto the conductors. The opposite charge of the two places creates a force. Where there is force, there must be energy. When we disconnect the source, the charge remains in the conductors, since they are attracted to each other.

Intuition

Capacitance is sort of like a spring, where is analogous to the spring constant. As charge builds up, the spring becomes more and more compressed.

Calculating Capacitance

Steps

  1. Choose an appropriate coordinate system
  2. Assume charge on one conductor and on the other
  3. Calculate the E-field (for ECE 106, usually Gauss's law)
  4. Find the potential difference
  5. Calculate capacitance

Usually will be in terms of some , and the will cancel out.

Putting Dielectrics Between Capacitors

See Dielectrics

The E-field of the dipoles superpose the original E-field, reducing the net field

Important

The charge stays the same, but the E-field is reduced. This causes lower potential and higher capacitance.

From Gauss's Law, we have , and from potential difference, we have . So we have:

Equation

Moving a Charge in a Capacitor

If we move a charge from the negative to positive place of a capacitor, what happens?

There's a potential difference of between the places. Since potential is work per unit charge we can write potential as a differential equation:
We also know the equation for capacitance is .
Combining these equations, we get . All we have to do now is integrate both sides:

Equation

The work to charge a capacitor is:

which is just the kinetic energy formula.

Energy Density of Electric Field

We can think of the energy stored in a capacitor as the energy being stored in the E-field itself.

Suppose we have a parallel plate capacitor. We have capacitance , given as , and the potential difference across the places at . Thus the energy stored in a capacitor is:

Equation

Note is the volume of the capacitor . If we divide by , we get the energy density. Thus the energy density is given as:

Equation

Generally:

Equation