Can electric fields penetrate a plasma in the presence of a magnetic field? Generally, electric fields can't penetrate plasmas, and this is known as debye shielding. But when I was thinking about the E cross B drift, this hit me.


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*The average motion of a particle along the perpendicular to the magnetic field is zero. By this I mean't to say particles do not drift perpendicular to the field lines. But for debye shielding to come into play, the particles must move in the perpendicular direction if the electric field is perpendicular to the magnetic field. So debye shielding doesn't work here.

*An other way I thought about it is as follows: Assuming the magnetic field and electric fields (which are perpendicular to each other) are already present, a plasma is created. Now, the particles, don't follow the electric field lines but drift perpendicular to both the magnetic field and electric field (according to the E cross B drift.). Since the particles drift and do not follow the electric field lines, debye shielding does not happen
But I am pretty skeptical about this, Is this true?
Thanks
 A: Debye shielding occurs when the more mobile electrons are free to move to screen the rest of the plasma from an electric field.
Particles in a magnetized plasma gyrate about the magnetic field with a gyro-radius of $r=mv/qB$. Drifts in the center point of the motion, such as the $\vec{E}\times\vec{B}$ drift, happen over many gyro-radii - so the drift is not a smooth motion as you describe. The motion is more like a cycloid - imagine the particle is a point on the rim/outside of a wheel; the "drift" is like the motion of the center of the wheel. Therefore, fast electrons with large gyro-radii can still effectively move to screen the electric field (imagine what happens in the limit $v\rightarrow\infty$).
Furthermore, the electrons are perfectly free to move along the magnetic field lines (in a helix) to screen the electric field, so if the $\vec{E}$ and $\vec{B}$ fields are not perfectly orthogonal, some screening can occur. For example, if there is some component of electric field along $\vec{B}$, let's call it $\vec{E}_{\parallel}$, particles would feel a force $q\vec{E}_{\parallel}$ along the magnetic field. So, while there are corrections to the standard picture of Debye shielding, it still occurs.
Debye shielding is a concept related to Langmuir probes. And those are frequently used to diagnose magnetized plasmas, such as in magnetrons or magnetic fusion devices, albeit with the magnetic field corrections.
Let me give another example: every plasma experiment ever conducted has been in some small magnetic field, whether the Earth's or the galaxy's. Debye shielding is still regularly observed. This is because in such a weak field almost every gyro-radius is so large that the particle motion is in a straight line and therefore is indistinguishable from the non-magnetized case.
A: As you said, Debye screening prevents electric field from affecting the plasma over distances larger than the Debye length. You are correct that it becomes more complicated when you add an external magnetic fields, basically you now enter the field of magnetohydrodynamics. 
Let's first assume we have a charged particle (with the charge $q$) which is not in motion (highly unlikely) and is placed in a magnetic and electric field. The Lorentz force, $\mathbf{F} = q\mathbf{E} + q\mathbf{v}\times\mathbf{B}$ will act on the particle and as $|\mathbf{v}|=0$, it will only be attracted by a the $\mathbf{E}$-field. As soon as it starts moving, the $\mathbf{B}$ will act on it and it will move perpendicular to both. 
The same happens in a plasma, so if you have an electric field the plasma particles will experience the $\mathbf{E}\times\mathbf{B}$ drift, exactly as you said. They will not cancel out the $\mathbf{E}$-field, unless it is somewhat parallel to the $\mathbf{B}$-field, as indicated by @Valentin Aslanyan. Note that this actually happens in stellarators and tokamaks, you do have electric fields there, generated by plasma processes (transport processes for example) that will influence the plasma particles' motion. One way to cancel these fields is due to motion parallel to the field lines, e.g. Pfirsch-Schlüter currents (or additional transport processes).
And then, as @Valentin Aslanyan said, if the magnetic field is so low and thus the gyroradius so large (larger than the experiment's typical spatial dimension), you can neglect it. You call a plasma basically magnetized if the $\mathbf{B}$-field is large enough to alter the particle trajectories (and/or if the collision frequency is smaller than the frequency of the gyration around the field line).
A: Paul Bellan's textbook, Fundamentals of Plasma Physics, Cambridge Press 2006, discusses particle motion on fields in plasma extensively.  He notes [3.5.2] that in a more general situation, "a charged particle will gyrate around B (local mag field line direction], stream parallel to B, have ExB drifts across B, and may also have force based drifts.  This is under the assumption that "these various motions are well separated and related to the requirement that the [E and B and other F fields] fields vary slowly and also to the concept of adiabatic invariance".
Bellan gets into drift equations quite extensively in Ch. 3, and leads into the concept of "magnetic mirrors" in Section 3.5.6.  Understanding magnetic mirrors and that they may be able to act as trajectory guides for particle motion, even at relativistic velocities, where synchrotron radiation can cause emission of pulsed light trains may give insight into Healey and Peratt's paper on the theory and lab experiments regarding the origin of very fast light pulses from pulsars, that contradicts the present view, that posits that a pulsar mass actually physically spins at the pulse rate in the "lighthouse" analogy. {Ref: "Radiation Properties of Pulsar Magnetospheres: Observation, Theory and Experiment", Kevin Healey (VLA Ops Center, NRAO), Anthony Peratt (Physics Division, Los Alamos National Laboratory), 1995.
