Formula sheet: Electrostatic Potential and Capacitance

1.1 Potential and potential energy

The potential at a point measures the work that an external agent must do to carry a unit positive charge there from infinity, moving it slowly. The electrostatic force is conservative, so that work is fixed by the endpoints alone, which allows every point in space to carry a single potential value. Being a scalar, potential is easier to add than the field.

The potential difference between two points is the work per unit charge in moving between them. The field always points from higher potential to lower potential, and the size of the field is the rate at which potential falls with distance. For a point charge the potential falls as one over r, not as one over r squared like the field.

For a system of charges, add the scalar potentials of the individual charges at the point of interest. There is no vector addition in potential problems, and that is its main practical advantage.

1.2 Capacitance and stored energy

A capacitor stores charge on two conductors held at a potential difference. The capacitance depends only on the geometry and the medium between the conductors, never on the charge placed on it. A parallel plate capacitor improves its capacitance by increasing the plate area, reducing the separation or inserting a dielectric.

The standard results for a parallel plate capacitor are the most used expressions in the chapter. A dielectric of constant K multiplies the capacitance by K while the geometry stays the same. Combinations follow the opposite rules to resistors: capacitors in parallel add directly, and capacitors in series add as reciprocals.

1.3 Units that must be consistent

Potential is measured in volt, potential energy in joule, capacitance in farad, charge in coulomb and the field in V m^-1 or N C^-1. Convert microfarad and picofarad before substituting, and remember that a dielectric constant is a pure number.