Can Coulomb's law ever be used with moving charges?

Coulomb's law is typically described as being for two stationary, isolated charges.

Coulomb's law, or Coulomb's inverse-square law, is a law of physics that describes force interacting between static electrically charged particles.

"Coulomb's law", Wikipedia

Even though it's primarily intended for stationary charges, can Coulomb's law ever be used with moving charges? If so, why?

I'm looking for an intuitive explanation, one concerning more of physics and less of mathematics.

• It also works in the "quasi-static" regime where the charges have velocities much less than the speed of light. If $v\approx c$, then one must use relativistic electrodynamics. May 2, 2018 at 17:50
• – Nat
May 6, 2018 at 23:25

1 Answer

Coulomb's law is not equally applicable to charged particles in motion. The reason is that moving charged particles give rise to magnetic fields in addition to electric fields. In a system of moving charged particles, to calculate the net force on any particular charged particle we use a different force law (the Lorentz force):

$$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B}),$$

where $q$ is the particle's charge and $\vec{v}$ is its velocity, and $\vec{E}$ and $\vec{B}$ are the (vector) values of the electric and magnetic fields due to all the other moving charged particles at the location of the charge in question. This is the force at one instant of time (think of a single frame in a motion picture). Because all the charged particles are moving, $\vec{E} \text{ and }\vec{B}$ will be continuously changing.