# How does a stationary charged particle kept in a magnetic field appear to a stationary and a moving observer?

Suppose a charged particle kept in magnetic field. There are two observers watching it. If one of the observer suddenly starts moving then I know that it will appear to him that the charged particle is moving in the opposite direction and he is going to see the particle move around in a loop.

But what about the stationary observer he is still at rest and for him the particle is going to be at rest forever not the slightest affected by the magnetic field.

So what is the particle actually doing?

Any help is appreciated :)

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Suppose for completeness that the charge $$q$$ is in a magnetic field pointing in the y direction $$\vec B = (0,B,0)$$. The force on $$q$$ is given by $$\vec F = q \vec E + q \vec v \times \vec B$$. Since $$\vec E = (0,0,0)$$ and $$\vec v = (0,0,0)$$ we obtain that $$\vec F = (0,0,0)$$ in the original frame.
Now, consider a reference frame where the charge is moving at velocity $$\vec v'=(v',0,0)$$ with axes aligned parallel to the original frame (I will use primed quantities to represent things in this frame and units where c=1). The transformation of the field is well known and, in this case, results in $$\vec E' = \gamma (\vec v \times \vec B) = (0,0,-\gamma \ v' \ B)$$ and $$\vec B' = \gamma \vec B = (0,\gamma \ B,0)$$ so $$\vec F' = q \vec E' + q \vec v' \times \vec B'= (0,0,-q \ \gamma \ v' \ B)+(0,0,q \ \gamma \ v' \ B)=(0,0,0)$$