How to Find Capacitance

The capacitance of an object is the ability for it to hold electric charge. Capacitance is related to charge and electric potential by the relation Assuming the existence of an electric charge finding the capacitance of an object will therefore require the calculation of the electric field and its associated potential before evaluating.

The capacitors in this article all exhibit some form of symmetry, which allows for the electric field to easily be evaluated. (Almost all problems using Gauss's law require some form of symmetry for the integrals to be evaluated analytically.)

EditPreliminaries

• Gauss's law: the electric flux penetrating a surface bounding a volume is equal to the charge enclosed within the volume.
• Electric potential is related to the electric field by the line integral below. In electrostatics, the electric field is defined to be conservative, so the field exhibits path-independence. Therefore, evaluation will only depend on values at the boundaries, and not on the specific path taken. The curve is being taken from the point of 0 potential to some other point.

EditSteps

EditParallel-plate Capacitor

1. Find the electric field. The principle of superposition allows us to find the electric field of only one of the plates. In this case, we use a Gaussian pillbox. This pillbox has a face parallel to the plate with area Let's define an areal charge density with units of charge per unit area. Because the electric field is perpendicular to the surface, only the ends of the pillbox will contribute to it.
   • Since we are dealing with two parallel plates that have charge densities and we recognize that when we consider the negatively charged plate only, the electric field points towards it. By the superposition principle, the electric field between the plates is doubled. As we are only interested in the magnitude of the electric field, its direction is ignored.
2. Find the electric potential. We assume that the plates are a distance apart. Then, we recall that the electric field is conservative, so it exhibits path-independence, again allowing for easy evaluation of the integral. Potential is always evaluated from high to low potential - in other words, in the direction of the electric field. Furthermore, when we find potential, we are only interested in finding its magnitude.
3. Find the capacitance. Using the relation as outlined earlier, we evaluate the capacitance of a parallel-plate capacitor below, where is found to be independent of both and

EditCylindrical Capacitor

1. Find the electric field. We use a cylindrical Gaussian surface with radius between the two cylindrical conductors to determine electric field, where the inner cylinder has radius the outer cylinder has radius both cylinders have length and As before, the electric field is perpendicular to the plates, so the tops of the Gaussian surface do not contribute. Only the side with area does.
   • The electric field is pointing radially from the center of the cylinders.
2. Find the potential. We are evaluating from to
3. Find the capacitance. The charges cancel out, and once again, the capacitance is independent of charge.

EditSpherical Capacitor

1. Find the electric field. We use a spherical Gaussian surface with radius between the two spherical conductors to determine electric field, where the inner sphere has radius the outer sphere has radius and The area of the Gaussian sphere is
   • This is the same electric field as that of a point charge. The field points radially from the center of the spheres.
2. Find the potential. We are integrating from to
3. Find the capacitance. The expression can be simplified into

EditTips

• It is interesting to consider the capacitance of an isolated spherical conductor. In this case, so the potential is evaluated as below instead.
  • Then, the capacitance of this isolated sphere evaluates to