Relativistic Induced Magnetic Field in particle's frame Suppose there is a constant magnetic field: $\vec{\mathbf{B}} = B \hat{z}$. 
A charged particle orbits that magnetic field perpendicular to the magnetic field, and induces a magnetic field in the opposite sense to that magnetic field. In the particle's reference frame, the magnetic fields cancel out on a small scale (right??). 
Is it true, relativistically, that a charged particle will experience no external magnetic field? Either way, what are the consequences of this result? 
Thanks! 
 A: You are correct that the gyrating electron will induce a magnetic field opposite to the applied electric field. But this induced field is not large enough to cancel out the external field, not even in the vicinity of the gyrating electron. 
It is also true that transforming into a moving reference frame will also transform electric fields into magnetic fields and vice versa. Depending on the relationship between the electric and magnetic field it is possible to transform one of the two away. 
If $E>cB$ then you can find a reference frame in which the magnetic field is zero. If $E<cB$ you can find a reference frame in which the electric field is zero. In your case, there is no electric field. This means that there will be no frame in which the magnetic field vanishes.
What does this mean for the force seen by the electron in its frame of rest? When you move into the frame of the electron there will be both a magnetic field and an electric field. But, because the electron is not moving in this frame, there is no $v\times B$ force acting on the electron due to the magnetic field. Because $v=0$, not because the magnetic field vanishes. The acceleration that the electron is experiencing is only due to the electric field in the reference frame of the electron. 
A: Under the influence of a guide field, as in your example, a single charged particle will gyrate in a way as to produce a current to counter the guide field.  However, the gyration of the particle will NOT, in general, produce a large enough current to cancel the guide field.
It is possible to transform into a reference frame where either the electric or magnetic fields are zero, but only under certain circumstances.  The expressions for the Lorentz transformation of electromagnetic fields can be found here.
Maxwell's and Einstein's combined results showed that the electric and magnetic fields are different manifestations of effectively the same thing.  I am being somewhat careless with that statement, but the point is that one can "convert" an electric field into a magnetic field simply by transforming into the proper moving frame of reference.
