respect an inertial observer O, I have an object of weight m and charge $q$. It is in a electric field $E=(E,0,0)$ constant and a magnetic field $B=(0,0,B)$ constant. I have another observer o' that moves along the y axis with speed $(0,-E/B,0)$ I have to rewrite the dynamics equation respected o' observer and show that it is a cicular uniform motion.
I think the dynamics equation of the observer o' is the same than the other because his movement has acceleration zero but I can't prove is circular.
Okay , then.In o' frame $$\vec v=-E/B \hat j $$ $$\vec B= B_0 \hat k $$ $$\vec E=E_0 \hat i$$ $$\vec F_B=q(\vec v \hat j \times \vec B \hat k)=qvB \hat i=-qE \hat i$$ $$\vec F_E=qE \hat i$$
So, $$\sum\vec F=0$$ So, its same as in frame of observer O as no extra force comes into scene due to relative speed.
Now, if we observe the scenario in O frame ,
$$\vec F_E=qE \hat i$$ So, $$\vec v=qEt \hat i$$ $$\vec F_B=q(\vec v\times \vec B)=q(qEtB)\ \perp \hat v\ in\ X-Y \ plane$$ Now as $V$ and $F_b$ are $\perp$ then the motion is a circular due to $F_B$ but the drag due to $\vec E$ will distrott the circular motion and particle will move in a complicated path.
A very-very$\times \infty$ rough diagram of path:
The radius goes on increasing due to increasing speed and particle is dragged to +ve direction of X-axis.