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Mathematics for Physics

27 Capacitance

A geometric property of an arrangement of conductors that is used to measure how much charge the conductors store per unit potential difference.

The definition of capacitance. Capacitance is the stored charge divided by the potential difference between the conductors. This principle is used to relate \(Q\) and \(V\) for a capacitor.

The definition of capacitance is

\[ C = \dfrac{Q}{V} \]

where

  • \(C\) is the capacitance.
  • \(Q\) is the magnitude of the charge on each conductor.
  • \(V\) is the potential difference.

The parallel-plate formula. For a parallel-plate capacitor the capacitance is proportional to the area of the plates and inversely proportional to their separation. This principle is used to design the simplest capacitors.

The parallel-plate capacitance is

\[ C = \epsilon_{0}\dfrac{A}{d} \]

where

  • \(C\) is the capacitance.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(A\) is the area of one plate.
  • \(d\) is the separation of the plates.

The stored energy of a capacitor. The energy stored in a capacitor is \(\dfrac{1}{2}CV^{2}\). This principle is used to compute the work required to charge the conductors.

The capacitor energy is

\[ U = \dfrac{1}{2}CV^{2} \]

where

  • \(U\) is the stored energy.
  • \(C\) is the capacitance.
  • \(V\) is the potential difference.