# Validity of application of Coulomb's Law with moving charge?

Hi there,
I'm trying to brush up on some of my E&M and am comparing treatments of Coulomb's Law in introductory calculus-based texts with higher level material. My understanding was that Coulomb's Law (and by extension, calculation of the electrostatic force on a charged particle using F = qE) was really strictly only valid when the charges were perfectly static. My understanding is that this is due to the fact that magnetic forces arise when there is motion of charges and as well that there are retardation effects if electric fields vary rapidly. I've run into two problems in these introductory texts that have me questioning some of my understanding of electrostatics.

Problem #1

A common problem that is introduced in first-year textbooks is the case of a moving charge like an electron undergoing projectile motion while in a uniform E-field. This is usually treated by considering the moving charge, working out the net force from the constant electric field and calculating characteristics of the trajectory under similar considerations to projectile motion. My question is concerning the validity of using the F = qE equation in this context.

This is clearly not a purely static situation, but my supposition is that it works because there is only the one charge moving around and all of the other source charges remain fixed so from the perspective of the mobile charge it's a "static" situation. As well, from the perspective of the one mobile charge, the other charges would not be in motion, so there ought to be no magnetic fields acting on it, and thus the magnetic force can be neglected. Is this interpretation correct or are there additional factors that are being papered over at an introductory level?

Problem #2

A second common type of problem is introduced along with the electrical potential energy for point charges. It involves two charged particles (such as a proton and an electron) that are released from rest and are allowed to accelerate towards each other and the problem asks to determine the speed of the particles when they are a certain distance apart. Here, magnetic forces surely cannot be neglected (since there are two moving charges), but is it still reasonable to solve this type of problem just using the concept of changes in electrical potential energy? Does this really give the correct answer even if magnetic effects are present? I would imagine that there would also have to be some loss of energy due to the production of radiation as the charges accelerate?

I would greatly appreciate if anyone out there could add any clarity to these questions and point out any misconceptions I have.

Thanks!

Alexander.

mfb
Mentor
My question is concerning the validity of using the F = qE equation in this context.
That is perfectly valid in the frame where the electric field is static.
From the perspective of the charge moving through the lab, the capacitor plates are moving and things are more complicated.

#2
Neglecting the magnetic field and assuming the electric field to perfectly follow the charges is a good approximation as long as the speed of the particles is slow compared to the speed of light. At a point where this is can become wrong you need quantum mechanics anyway.

• Alexander83
Thank you, mfb. Your answer makes good sense and helps a lot!

Cheers, Alex.