Does a moving magnetic mirror accelerate particles?

Question

I understand that a magnetic mirror can confine particles within it. Now, suppose I have a magnetic mirror created from two electromagnets :

If those two magnets start to accelerate towards the right (as in the diagram):

• In the frame of reference of the electromagnets, the particles would gain speed towards the left, and some should get confined, reflected and accelerated.

• In the frame of reference of the particles, some would get accelerated towards the right.

Are these assumptions, in a theoretical framework, correct?

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If so, would a moving magnetic mirror induce a current in a stationary wire (wire not moving and passing through the center of both electromagnets), because the protons shouldn't move and so only the electrons would get accelerated?

The question is theoretical and complex, so please do not get hung up on details like whether two electromagnets would create an efficient magnetic mirror... I hope my questions are detailed, enough; feel free to tell me if they aren't.

Answer 1

A changing magnetic field induces an electric field, which can change the energy of charged particles.

Consider the “turning point” for the charged particle, when its momentum is briefly entirely perpendicular to the non-uniform magnetic field. In a uniform field, the particle’s path would be a two-dimensional circle. The magnetic mirror effect is that the Lorentz forces conspire to steer this charged particle towards the direction of the weaker field. In the weak-field region, the particle will have substantially “straightened out” its path, with nearly all of its momentum parallel to the field. This process is time-symmetric, which explains why an incoming particle is reflected. If the magnetic field is constant, then the reflection is elastic, with the outgoing linearized momentum equal to the incoming momentum.

Now boost into a reference frame where the magnetic mirror is moving. In this reference frame, the incoming and outgoing charged particles have different momenta. The changing magnetic field has done work on them.

Your question about current in a wire confused me, because we don’t usually spend much time discussing wires parallel to the magnetic field. Your moving magnetic mirror would induce a current in a bulk conductor, but only in regions where the conductor is large enough to accommodate the gyroscopic paths of the moving charges. Compare to eddy-current braking, which can be eliminated by strategically cutting holes or slits in the conductor so that there are not eddy-current-sized paths for the conduction electrons to follow. A practical example of a “magnetic mirror force” is that a permanent magnet dropped near a superconductor will hover above it, rather than falling. (The hovering configuration is unstable, so the hovering magnet tends to tumble.) A small-diameter wire parallel to the field of a magnetic mirror would only experience a current in the very strongest parts of the changing field.

Note that, since charges of both signs are repelled by the strong-field region, the sign of the mirror-induced current will depend on the abundances, mobilities, and (effective) masses of the various charge carriers in the conductor. In a metallic conductor, an increasing magnetic field would repel the more mobile electrons, giving one sign of current. In an electrolyte with multiple mobile ion species, a clever experimenter might be able to exploit the relation between mass, mobility, and gyroradius to build an apparatus where the sign of the induced current can go either way.

Answer 2

For the case of a wire sliding through a mirror machine, it's probably useful to consider each of the high-field "mirror" sections separately. The electrons and positive ions are going to have very different gyroradii and confining forces, so we'll start by thinking about single-species plasmas.

• At the peak strength of the B field and in the constant, lowest-field regions, B is parallel to the direction of motion, so there's no F = q(E + v x B) Lorentz Force and thus no acceleration on the charge carriers.
• In the transition regions, there is a radial component to the magnetic field, perpendicular to the nominal direction of the charge carriers' axial travel. This induces an azimuthal acceleration, but can do no work on the charged particles - they maintain constant speed (in either frame), but change direction as dictated by the Lorentz force. Note that the positive and negative charges are deflected in opposite directions, yielding an azimuthal current. With this induced (azimuthal) current flow, there is interaction with the axial component of B, and the sign of the resultant deflection turns the charge carrier back in its original direction.
• The mirror reflection is charge-symmetric: both positive and negative charges are repelled from the strongest magnetic field, and the strength of this repulsive force is related to the charged particles' speed. This yields a useful concept of magnetic pressure, which can be compared to the charge carriers' thermal pressure. This comparison gives the effectiveness of the magnetic mirror and some intuition about the low-order net effects of the relative motion between the mirror and the conductor: There will be a depletion of charge carriers near the maximum field / magnetic mirror. Both positive and negative charge carriers will experience this local-depletion force. Your moving-mirror configuration will tend to trap the charges within in, with charges leaking in through the "front" mirror and leaking out through the "back" mirror as they are scattered by thermal & magnetic effects into the mirror's loss cone.
• The azimuthal current induced in the charge carriers will generate a magnetic field opposing the mirror gradient, weakening the mirror and widening its loss cone.
• Lattice constraints on the metal ions suggest you'll be doing work on the metal, squeezing it with the magnetic mirrors as the conductor passes through the high-field region.
• Collisions between positive and negative charges will yield resistive heating. This weakens the induced azimuthal current, but also randomizes individual particles' velocities, scattering some fraction of the "trapped" particles into the loss cone.
• The resistive losses will sap energy from the relative motion between the conductor and the magnets: The forward motion of the magnets will be converted to local resistive heating of the conductor. Also note that as metals' temperature increases, their resistivity usually increases as well.
• Depending on the material of your conductor, ferromagnetic and paramagnetic effects will change the strength (and possibly sign) of the net pressure acting on the metal, since the isolated protons and electrons considered above are diamagnetic. However, the overall picture should remain the same: The motion will induce local azimuthal currents and resistive losses that heat the metal and remove energy from the magnets' / conductors' net motion.
• Geometric constraints that interfere with the free-space gyroradii will similarly weaken the diamagnetism, since they cause a reversal in sign between the directions of the micro- and macro-scale orbits. Notably though, geometric constraints don't prevent bulk currents or resistive losses.

Strictly speaking, your mirror-moving-along-a-conductor scheme will induce currents, but those will be localized to the areas with substantial magnetic gradients and subject to substantial resistive losses.