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Mathematics for Physics

33 Coulomb Force

The electrostatic force between two stationary point charges that is used to compute the push or pull of one charge on another, where a point charge is a charge treated as concentrated at a single location.

Coulomb’s inverse-square law. The force is proportional to the product of the two charges and falls as the inverse square of their separation. This principle is used to compute the electrostatic force between two isolated charges at rest.

Coulomb’s law is

\[ \mathbf{F} = \dfrac{1}{4\pi\epsilon_{0}}\dfrac{q_{1}q_{2}}{r^{2}}\hat{\mathbf{r}} \]

where

  • \(\mathbf{F}\) is the force on \(q_{2}\).
  • \(q_{1}\) and \(q_{2}\) are the two charges.
  • \(r\) is the separation.
  • \(\hat{\mathbf{r}}\) is the unit vector from \(q_{1}\) toward \(q_{2}\).
  • \(\epsilon_{0}\) is the permittivity of free space.

The sign rule for attraction and repulsion. The force is repulsive when the charges have the same sign and attractive when the charges have opposite signs. This principle is used to determine the direction of \(\mathbf{F}\).

Newton’s third law for a charge pair. The force on \(q_{1}\) due to \(q_{2}\) is equal and opposite to the force on \(q_{2}\) due to \(q_{1}\). This principle is used to treat the pair as a Newton’s-third-law pair.

Newton’s third law for the Coulomb pair is

\[ \mathbf{F}_{12} = -\mathbf{F}_{21} \]

where

  • \(\mathbf{F}_{12}\) is the force on charge 1 by charge 2.
  • \(\mathbf{F}_{21}\) is the force on charge 2 by charge 1.

Superposition of Coulomb forces. The net Coulomb force on one charge is the vector sum of the forces from all the others. This principle is used to compute the force in a many-charge collection.

The superposition of Coulomb forces is

\[ \mathbf{F}_{1} = \sum_{i\neq 1}\mathbf{F}_{1i} \]

where

  • \(\mathbf{F}_{1}\) is the net force on charge 1.
  • \(\mathbf{F}_{1i}\) is the Coulomb force on charge 1 by charge \(i\).