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We have been taught that the force on a moving charge due to magnetic field is: $$\vec{F} = q(\vec{v} \times \vec{B})$$ When I asked my professor about its source, he said that it was derived from mere observations. But from my knowledge, generally formulas derived from observations contain some proportionality constant which is not seen in this formula.

I looked up in many different books and on web but I did not find any derivation of it. From the derivation, I want to know that why does a moving charge experience this force in magnetic field.

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This formula is derived long after almost all of the other laws of electromagnetics were derived. It was actually derived and not just from mere observation (as far as I know).

Lorentz found that the world is not as we think it is and a moving person sees a phenomenon differently than a stationary person.

Lets understand that :

Say you have a charge that is moving with a constant velocity in a magnetic field. You would see it experiencing a force that makes it go in a circle. Now imagine you are also moving with the particle so that for you, the charge is stationary. So it should seem that the charge is not experiencing any force and thus continue to move with you without changing direction or making circles.

So how does a same phenomena appear different to someone moving? to make it clear, you may imagine two people - one stationary and other moving with the charge. Imagine the weirdness of one seeing the charge moving in circle and the other seeing it stationary (ie:. continuing to. move straight).

Lorentz was working on a completely different thing - trying to explain the constant speed of light in michleson-morely experiment and he came up with a fact that lengths contract when we are in motion. His conclusions were the lorentz transformations, which made your measurements different if you are moving. Don't bother about these now.

Now, putting these effects into electromagnetics, another weird thing came out which could explain why the first weird thing would not happen. The conclusions were : When you move, the magnetic fields are not the same as before. They vary such that for the charge (or equivalently for an observer that is moving with the charge) , it feels like an electric field.

THE PROOF

mathematical statements seem unnecessary here. The idea is : If we imagine a stationary charge experiencing some Electric Field and then you move into a moving person's perspective, you would see the electroc field showing completely different properties but the charge moving just as it should be(same in both frames). The new force that the charge experience becomes the lorentz force.

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  • $\begingroup$ ..in THE PROOF : Could Force Law of stationary charge $\mathbf f \boldsymbol{=} q \mathbf E$ be derived from Maxwell Equations ??? $\endgroup$ – Frobenius May 17 '20 at 13:35
  • $\begingroup$ Not actually... Maxwell's equations cannot derive this. F = q E is by definition. We define E as force on a unit charge. And force is derived from Coulomb's laws. But going backwards is also possible. You can derive E from maxwell's equations $\endgroup$ – Rishab Navaneet May 18 '20 at 2:00
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There is no proportionality constant because the Lorentz force law provides the definition of $\mathbf B$.

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  • $\begingroup$ So you mean to say that the formula is derived from observations? $\endgroup$ – Arnav Mahajan May 17 '20 at 6:35
  • $\begingroup$ Yes. I agree with your professor. You can “derive” it from a Lagrangian, or in other ways, but where does that come from? $\endgroup$ – G. Smith May 17 '20 at 6:40
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    $\begingroup$ Ultimately the point of physics is to explain observations. We know of no reason why there has to be an electromagnetic field at all, and we can imagine universes where EM is very different (for example, highly nonlinear), so of course the equations for it, such as Maxwell’s equations and the Lorentz force law, aren’t true a priori. $\endgroup$ – G. Smith May 17 '20 at 6:52

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