# Force on a moving charge in relative frame

Consider a particle of charge $q$ moving at velocity, $v$, enters a magnetic field $B$ which is perpendicular to its motion.

Now consider two observers, one stationary and the other moving at the same velocity, $v$.

The stationary observer, notices a force, $\vec{F}= q(\vec{v} \times \vec{B})$ which deflects the charge and makes it go in a circle.

For the moving observer, the charge is stationary, so it observes no force. However, the magnetic field moves in (changing magnetic field?). I can not figure out what equation to apply. Any thoughts.

Note: I am just curious, this is not a homework question

## 1 Answer

The electric and magnetic fields transform like a second rank tensor not a four vector. I would suggest you look in Jackson classical Electrodynamics for the transformations. One observer will see a moving magnetic field which will have an electric field component. At low velocity this will just be interpreted as magnetic induction.

• It might be useful to paraphrase the relevant section of Jackson to show the case, rather than directing someone to a book that they may not possess. – Kyle Kanos Apr 27 '15 at 1:43