# Force on a charged particle inside magnetic field

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?

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.

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.

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.

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

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.

A classic example of a pimue decay

see this for how the centrifugal balances the magnetic force.