# Coulomb's law (electrostatic force)

I have read about the electrostatic force so it is said that charges must be at rest, but in some exercises, the electrostatic force between the electron and the proton in an atom is calculated. These exercises themselves say that the electron has a speed $$v$$ around the proton.

According to this situation, I would expect that the Coulomb's law must not be applied here because the electron is moving around the proton but the exercise uses the Coulomb's law to calculate the force acting on the electron.

I have read that the electrostatic force can be also applied to the interaction among atoms in molecules and among atoms and molecules. In this last cases shall we assume the relative speed of each atom relative to another is zero? I say this because the molecules could move.

Electrostatics (and magnetostatics) remain valid without modification for low velocities, which means $$v/c \ll 1$$.

When this is not the case one must use (special) relativity and account for the transformations of the electric and magnetic fields under Lorentz transformations.

• One can just solve Maxwell equations (like Liénard and Wiechert) and don't have to worry about any of that post 1905 stuff... But of course you're right. I just want to mention that these consdierations are what led to Einsteins idea. – TheoreticalMinimum Mar 12 at 15:47
• If we think about the Bohr's model, the electron is moving around the proton with a low velocity. But, I guess, Coulomb's law is still valid also because the electron's charge is very small so that we can think to the electron as a test charge. Is it correct? – Angelo Giannuzzi Mar 12 at 16:34
• No. the size of the charge does not enter. Only the velocity compared to speed of light. – ZeroTheHero Mar 12 at 18:42
• And what about molecules? According to Halliday et al., the Coulomb's law also applies to the force among atoms in molecules. If the molecules are not at rest, can we apply anyway the Coulomb's law? – Angelo Giannuzzi Mar 12 at 19:30
• as long as velocities are $v/c\ll 1$. – ZeroTheHero Mar 12 at 19:45

In these calculations one is interested in the magnitude of the interaction, which is predicted correctly by applying Coloumb. For rigorous treatment of intermolecular and atomic interactions one must resort to quantum theory anyways.

• But according to Bohr's model, the electron is moving – Angelo Giannuzzi Mar 12 at 13:15
• Well yeah. As I implicitly said using Coloumbs Law here is wrong (as is Bohrs model). But it will give a rough estimate for the magnitude of the interaction. – TheoreticalMinimum Mar 12 at 13:52
• Ok, now it is clear. Thank you. – Angelo Giannuzzi Mar 12 at 13:55
• Check out the Liénard–Wiechert potential. But I think you will need to learn a lot of electrodynamics beforehand.. – TheoreticalMinimum Mar 12 at 15:13

The statement that Coulomb's Law is only for static charges is concerned with accuracy, but does not imply that the force no longer acts. A disturbance in an electric field moves outward at a finite speed (c), so the force on a second charge may not reflect that the first charge has moved. Or, if the second charge is moving, it may experience an additional force from any existing magnetic field.