# 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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## 1 Answer

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 $$\vec F' = q \vec E' + q \vec v' \times \vec B'= (0,0,-q \ \gamma \ v' \ B)+(0,0,q \ \gamma \ v' \ B)=(0,0,0)$$

Therefore, in both frames the force on the particle is 0. In the original frame it is 0 because there is no electric field (electric force = 0) and the charge is stationary (magnetic force = 0). In the frame where the charge is moving, there is a magnetic force which is balanced by an opposite electric force.

• in the first place thnx a lot Dale. but the truth i that i am a first year medical MBBS student, and i still keep on exploring physics just bcoz i find it so damn cool. So i am finding these equations tough to understand. Hence in laymans language its like in the moving frame the particle is going to even feel an electric force which wont let it move or else we would end up violating the Law conservation of energy – Alpa Patel May 3 '20 at 16:47
• @Alpa Patel yes, sounds like you got it! – Dale May 3 '20 at 23:57