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I have just started learning electrodynamics. And I came upon this expression telling force on a charged particle inside a magnetic field.

$$\vec F =q( \vec v ×\vec B) $$

where, $F$ = Force on that charged particle.

$v$ = velocity of that particle.

$B$ = magnetic field.

My question is that I know magnetic field and electric filed are frame dependent, so here in this formula velocity ($v$) is respect to which frame?

Thanks in advance.

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4 Answers 4

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This formula $\vec F =q( \vec v ×\vec B) $ is valid only for the frame where $\vec E=0$.

The actual formula for Lorentz Force is :

$$\vec F =q( \vec v ×\vec B +\vec E) $$

and this is valid for any frame.

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This formula (Lorentz force) is valid in any inertial frame. This means Lorentz force depends on inertial frame, it is not the same in all frames.

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It is the frame in which the charged particle's velocity is $v$ and in which the magnetic field is $\vec{B}$.

Both are transformed in other frames, and in general there will also be an additional force $q\vec{E'}$ in some other frame, because the B-field will transform into the sum of a new B-field and an electric field.

Note further that the force will not be the same in another frame.

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This is true , and it is classical electrodynamics. The definition

magfor

implies the frame where the particle is moving with velocity v, and there exists a static Magnetic field, the laboratory system.

(I do not know how to get vectors either)

See these real particles turning in the magnetic field in a bubble chamber picture.

pion

A classic example of a pimue decay

see this for how the centrifugal balances the magnetic force.

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