Magnetic force between two point charges

Question

I tried to derive the magnetic force between two point-charges for iterative computation. Starting out with Lorentz force and Biot–Savart law for a point charge. $$ \vec F = q_2( - \Delta \vec{v} \times \vec{B})$$ $$\vec B = (\vec \Delta v \times \vec \Delta x) ( \frac {q_1}{ ||\Delta \vec x||^3}\frac{\mu_0}{4\pi}) $$ And got this for the answer by direct subtitution: $$ \vec F = q_2(-\Delta \vec{v} \times (\vec \Delta v \times \vec \Delta x) ( \frac {q_1}{ ||\Delta \vec x||^3}\frac{\mu_0}{4\pi}))$$

It does not seem to be right for several reasons.

1. It seems as if electron would be attracted to the nucleus by both magnetic and electrostatic force. Considering hydrogen atom.
2. It is possible to show that between non-moving particle and a non-moving wire with current in it should exist magnetic force. (Magnetic force acts between charge carriers in the wire and the point-charge. Non-moving particles in the wire do not have influence on the magnetic force)

Where have I gone wrong and how to find the correct expression for the magnetic force between two point-charges? Does the equation hold if $ \Delta v << c$? If this equation proves to be wrong, what would be the correct approach?

Answer 1

It appears that you have used the Biot-Savart law which assumes no static electric field. Instead, you should use the Lorentz force law, $$ F =q(E +Δv × B),$$ which accounts for both magnetic and electrostatic interactions.

Answer 2

For calculating force between a point charge and a wire carrying current:

Force cannot be zero if the test charge is stationary with respect to the positive charges ( atomic lattices). Reason is the relative motion between the test charge and the free electrons moving with Drift velocity. However the force shall be extremely small and undetectable due to drift velocity being very less.

Force can mathematically be zero only if the test charge is stationary with respect to inertial frame S0 such that the electrons and the nucleii appear to move in opposite directions with equal speeds. This situation of force = 0 can be easily derived by understanding symmetry and Gallilean relativity. In other words if the test charge moves at a speed = half of Drift speed of electrons in the direction of the electron drift then mathematically and physically the electrodynamic force on it shall be NIL.

Thanks and Regards Suharto Sengupta