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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?

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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.

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