Electrostatics Summary

The SI unit of charge is the coulomb, and the fundamental unit of charge is e = 1.60 x 10⁻¹⁹ C.

Coulomb’s law gives the magnitude of the electrostatic force F between two charges q_1 and q_2 whose centers are separated by a distance r: F = k (q_1 q_2 / r²).  The force always points along the line connecting the centers of the two charges.

Coulomb’s constant or the electrostatic constant k is: k = 1/(4πε_0) = 8.99 x 10⁹ (N · m²) / C², where the permittivity of free space ε_0 is: ε_0 = 8.85 x 10e⁻¹² C² / (N · m²).

Every electric charge sets up a surrounding electric field.  The magnitude of the electric field can be calculated: E = F / q_0 = kq / r².

The direction of the electric field vector is the direction that a positive test charge +q_0 would move.

A collection of charges will exert a net electric field at a point in space equal to the vector sum of all electric fields: E_total = E_q1 + E_q2 + E_q3 + …

The electric potential energy is the amount of work necessary to bring a charge from infinity to that point and is: U = k qQ / r.

Opposite charges will have negative potential energy, and this energy is increasingly negative as the charges are brought closer together.

Electric potential is the ratio of the magnitude of a charge’s electric potential energy to the magnitude of the charge itself: V = W/q = kQ/r.

The potential difference, known as the voltage, between points a and b is: V_b - V_a = W_ab / q_0.

Positive charges move spontaneously from high voltage to low voltage, while negative charges move from low voltage to high voltage.

An equipotential line is one where the potential at every point is the same, such that no work is done when moving a test charge q_0 from one point to another on the line.

The electric dipole, which results from two equal and opposite charges being separated by a small distance d, can be transient or permanent.

The electric potential P of the dipole is the scalar sum of the potentials due to each charge: V = kq (r_2 - r_1) / (r_1 r_2) ≈ (kqd / r²) cos θ

The product qd is the dipole moment p with SI units C · m, such that V = (kp/ r²) cos θ.

The perpendicular bisector of the dipole is the plane that lies halfway between +q and -q, where the electric potential is zero.  The electric field along the perpendicular bisector is: E = kp / r³.

The force applied to a dipole axis from the electric field generates torque and causes the dipole axis to rotate: τ = pE sin θ.