A How exactly is energy transferred in grad(B) drifts?

[TheCanadian]

When considering an ionized medium (e.g. plasma) in nonuniform magnetic field, $\vec{B}$, there will be a drift velocity for ions that goes in the direction of $\frac{1}{q} \vec{B}\times\vec{\nabla} B$: https://en.wikipedia.org/wiki/Guiding_center#Grad-B_drift

Although this implies the ion will be increasing its kinetic energy when entering a nonuniform magnetic field, for example if it's initially traveling perpendicular to this drift direction. Thus I am just curious as to where exactly energy is coming from and the mechanism by which this is occurring. Is the magnetic potential energy decreasing (as there is a gradient in the field) and being exchanged with the particle, thus leading to energy conservation? Is there a particular force typically attributed to this process? And in terms of photons being energy carriers (and the field being represented by creation/annihilation operators), is there an intuitive understanding as to how photons could mediate this process of producing a drift velocity on an ion when entering a magnetic field with non-zero gradient?

 

[Khashishi]

Assuming there is no E field, the kinetic energy of the ion doesn't change as it gyrates around the field lines. The velocity is moving in a circle, but the speed stays the same. What do you mean by "magnetic potential energy"?

 

[TheCanadian]

Assuming there is no E field, the kinetic energy of the ion doesn't change as it gyrates around the field lines. The velocity is moving in a circle, but the speed stays the same. What do you mean by "magnetic potential energy"?

The gyration still exists similar to the case of a uniform field, but there is now a new drift for the guiding centre of the ion itself which does result in an additional velocity to the particle (thus changing its kinetic energy). The magnetic potential energy I am referencing is that which is stored in the field: https://en.wikipedia.org/wiki/Magnetic_energy

 

[jartsa]

Are you familiar with this device:
https://en.wikipedia.org/wiki/Magnetic_mirror

That device is just a bunch of current loops, so hopefully we only need to understand one current loop bouncing off another current loop.

 

[jartsa]

Are you familiar with this device:
https://en.wikipedia.org/wiki/Magnetic_mirror

That device is just a bunch of current loops, so hopefully we only need to understand one current loop bouncing off another current loop.

So, if a small conducting loop is moving in a non-uniform magnetic field following the field lines as closely as possible, then every part of the wire is cutting field lines, because the field lines diverge. And an EMF is induced into the loop.

If a gyro-rotating particle is moving in a non-uniform magnetic field following the field lines as closely as possible, then every part of the gyro-path is cutting field lines, and an Lorentz-force induced into the particle. The particle either loses drift-speed and gains gyrating speed, or it gains drift-speed and loses gyrating speed.

By drift-speed I mean the speed that is not the gyrating speed, but the other one - speed along field lines.

Can you guess what gyro-path means?

If the non-uniformity of the magnetic field is such that a wire loop's left side is in a weak uniform magnetic field, while its right side is in a strong uniform magnetic field, then the motion of the loop along the field lines does not cause any EMF at any part of the loop. ... That's all I have to say about that case.

 

[Khashishi]

The gyration still exists similar to the case of a uniform field, but there is now a new drift for the guiding centre of the ion itself which does result in an additional velocity to the particle (thus changing its kinetic energy). The magnetic potential energy I am referencing is that which is stored in the field: https://en.wikipedia.org/wiki/Magnetic_energy

The guiding center drift velocity is not an "additional" velocity. It comes a cycle average of the gyro orbit. All of the kinetic energy is accounted for in the gyro orbit.
To understand the grad B drift, consider that the gyro orbit radius depends on the magnetic field amplitude. Since the amplitude depends on position, you don't have circular orbits. If the magnetic field is higher on the left side, then the orbit is a distorted circle such that the right side covers a bigger arc than the left. The result is that the loop doesn't close on itself, and there is a small displacement after ever cycle.

The magnetic stored energy is not relevant to the grad B drift. There is another drift, the diagmagnetic drift, which does have an effect on the magnetic stored energy.

 

[TheCanadian]

The guiding center drift velocity is not an "additional" velocity. It comes a cycle average of the gyro orbit. All of the kinetic energy is accounted for in the gyro orbit.
To understand the grad B drift, consider that the gyro orbit radius depends on the magnetic field amplitude. Since the amplitude depends on position, you don't have circular orbits. If the magnetic field is higher on the left side, then the orbit is a distorted circle such that the right side covers a bigger arc than the left. The result is that the loop doesn't close on itself, and there is a small displacement after ever cycle.

The magnetic stored energy is not relevant to the grad B drift. There is another drift, the diagmagnetic drift, which does have an effect on the magnetic stored energy.

Perhaps I am missing something blatantly obvious and elementary, but in the derivation of this drift velocity due to the magnetic field's gradient, where is the (explicit or implied) constraint that the particle's kinetic energy is conserved? In the case of a uniform magnetic field, it is easy to see that the particle will be in a circular orbit where its speed remains constant (there is no potential difference), but that doesn't appear so obvious here as there is no longer simply uniform circular motion.

 

[Khashishi]

If you are only considering single particle motion (which is all that is needed to understand the grad-B drift), when E = 0, the Lorentz force doesn't do work, so the energy is conserved. In a magnetic mirror, there is a conversion between parallel kinetic energy and perpendicular kinetic energy, but the total energy doesn't change.

 

[TheCanadian]

If you are only considering single particle motion (which is all that is needed to understand the grad-B drift), when E = 0, the Lorentz force doesn't do work, so the energy is conserved. In a magnetic mirror, there is a conversion between parallel kinetic energy and perpendicular kinetic energy, but the total energy doesn't change.

Well I suppose I am questioning what exactly a $\vec{\nabla} B$ drift inherently means. From simply looking at the situation, one at first notices there is no net energy gain/loss for the particle as there is no electric field and the magnetic field conserves the particle's energy. But if someone turned the non-uniform magnetic field on and off, then the particle would switch from having a drift to not having a drift. I suppose I am curious as to how this particle has a drift in the first place when it simply enter the nonuniform magnetic field. What is acting on the particle for this motion to arise?

 

[TheCanadian]

So, if a small conducting loop is moving in a non-uniform magnetic field following the field lines as closely as possible, then every part of the wire is cutting field lines, because the field lines diverge. And an EMF is induced into the loop.

If a gyro-rotating particle is moving in a non-uniform magnetic field following the field lines as closely as possible, then every part of the gyro-path is cutting field lines, and an Lorentz-force induced into the particle. The particle either loses drift-speed and gains gyrating speed, or it gains drift-speed and loses gyrating speed.

By drift-speed I mean the speed that is not the gyrating speed, but the other one - speed along field lines.

Can you guess what gyro-path means?

If the non-uniformity of the magnetic field is such that a wire loop's left side is in a weak uniform magnetic field, while its right side is in a strong uniform magnetic field, then the motion of the loop along the field lines does not cause any EMF at any part of the loop. ... That's all I have to say about that case.

So the drift speed I suppose is not constant as it is typically stated, but increases as it begins drifting which results in the gyrating speed to decrease, but then continues to increase gyrating speed because the drift velocity is still perpendicular to the magnetic field. Thus there would be an almost cyclical balance between the drift speed and gyrating speed?

What exactly is happening if a particle enters a region with a nonuniform magnetic field and begins drifting in a different direction? What exactly are we attributing as the force to cause this drift in a particular direction that is not necessarily perpendicular to the particle's initial velocity?

 

[Khashishi]

Here's a picture to go along with my explanation in post #6.

 

[TheCanadian]

View attachment 211924
Here's a picture to go along with my explanation in post #6.

The picture and re-reading your post helped immensely. Thank you.

 

[jartsa]

What exactly is happening if a particle enters a region with a nonuniform magnetic field and begins drifting in a different direction? What exactly are we attributing as the force to cause this drift in a particular direction that is not necessarily perpendicular to the particle's initial velocity?

Well I like to think that a particle experiences an electro-motive force as the magnetic field around the particle is changing, and changing magnetic field is equivalent to an electric field. That's what happens in the magnetic mirror case.

Sometimes I like to think that a particle experiences a Lorentz-force as the the particle is cutting magnetic field lines, and cutting magnetic field lines is equivalent to being in an electric field. This is still the magnetic mirror case. One must forget that the particle gyro-rotates and consider only how the motion of the guiding center of the particle causes cutting of magnetic field lines.

By the way, earlier I called the motion of the guiding center along the magnetic field lines drifting, that was a very bad word choice. 'Motion' is a perfectly good word for that motion. And drifting means ... something different.Now, I would like to talk about torque, so let's pick two particles that gyro-rotate at same rate and are at opposite phase of rotation. That system may experience a torque for some time. The rotational energy of the system changes by this amount: torque * angle, where angle is the angle that the system rotated. Somewhere there has to be a reaction torque to that torque. Where ever that reaction torque is, there occurs a change of energy: torque * angle₂

 

[jartsa]

So the drift speed I suppose is not constant as it is typically stated, but increases as it begins drifting which results in the gyrating speed to decrease, but then continues to increase gyrating speed because the drift velocity is still perpendicular to the magnetic field. Thus there would be an almost cyclical balance between the drift speed and gyrating speed?

Sorry but I have been off topic all the time.

I haven't said anything about drifting yet. As far as I understand the above has nothing to do with drifting, as the above is a description about a particle inside a magnetic mirror machine, and the particle is just moving, gyrating and getting reflected from the mirrors.