Exploring Magnetic Mirrors in Fusion Reactors

[MoreKnowledge]

Hello! I hope that i am doing this right. This is my first post of a question. So, I've been reading about a design for a fusion reactor called a "Polywell". The Polywell, as I understand it, works by sending electron beams to the device's center which suck in protons/deuterium/tritium/etc. so that they collide with each other. Like with all proposed fusion reactors, minimizing energy loss is important. I am reading about how "magnetic mirrors" are used to contain electrons and/of protons so that they do not escape the system and take energy with them. At least this is how I've interpreted it. Is a magnetic mirror essentially a magnetic field that reflects particles that collide with it? Does it work kind of like throwing a ball against a wall, or like a cage fighter bouncing themself off the ropes? I know these analogies are not perfect, but do they capture the general concept? Also, in the future, should I ask shorter questions and/or provide less background information?

 

[mfb]

or like a cage fighter bouncing themself off the ropes?

Something like this. As usual, Wikipedia has an overview.

 

[the_wolfman]

Hello! I hope that i am doing this right. This is my first post of a question. So, I've been reading about a design for a fusion reactor called a "Polywell". The Polywell, as I understand it, works by sending electron beams to the device's center which suck in protons/deuterium/tritium/etc. so that they collide with each other. Like with all proposed fusion reactors, minimizing energy loss is important. I am reading about how "magnetic mirrors" are used to contain electrons and/of protons so that they do not escape the system and take energy with them. At least this is how I've interpreted it. Is a magnetic mirror essentially a magnetic field that reflects particles that collide with it? Does it work kind of like throwing a ball against a wall, or like a cage fighter bouncing themself off the ropes? I know these analogies are not perfect, but do they capture the general concept? Also, in the future, should I ask shorter questions and/or provide less background information?

Hi MoreKnowledge,

The amount of detail in your question is fine.

First the cartoon:
Charged particles gyrate around magnetic fields like beads on a string. Normally they are free to move along (parallel) the magnetic field, but their motion perpendicular to the magnetic field is deflected into circular orbits. Now imagine a loose bundle of finite length stings. Here the beads can easily slide on and off the ends of the stings. You can make it harder for the beads to slide of the ends by squeezing to ends of the strings together so that they're tightly packed. This akin to how the magnetic mirror works.

The actual physics behind the mirror effect is a little more complex. It involves the conservation of energy and the conservation of the magnetic moment. You can show that the quantity $\mu = \frac{m v_\perp}{2B}$ is conserved by a charge particle moving in a slowly varying magnetic field. Here $v_\perp$ is the particles velocity perpendicular to the magnetic field and $\mu$ is called the magnetic moment. Note that $v_\perp$ will increase as the as the magnetic field increases. Now the magnetic field can not do work on a charged particle. This is because the Lorentz force always acts perpendicular to velocity of the particle $\vec F_m = \vec v \times \vec B$ but work is defined as the force in the direction of the velocity $W = \int \vec F_m \cdot \vec v dt = \int \left(\vec v \times \vec B\right) \cdot \vec v dt = 0$. The fact that the work done on the particle is zero implies the the kinetic energy of the particle must be conserved. The total kinetic energy of the particle is the sum of the parallel and perpendicular kinetic energies $KE = \frac{1}{2}mv_\parallel^2 + \frac{1}{2}mv_\perp^2$. To conserver the kinetic energy the square of parallel velocity must decrease as the square perpendicular velocity increases. But the square of the parallel velocity cannot be negative. Thus the conservation of energy places a limit on the maximum allow allow perpendicular velocity.

We now have all the pieces together to understand the magnetic mirror. Let us consider the motion of a charged as it travels into a region of increasing magnetic field.
The particle has some initial perpendicular velocity and parallel velocity. As the particle travels "up" the field the strength of the field increases and the perpendicular velocity increases to conserve of the magnetic moment. As the perpendicular velocity increases the parallel velocity decreases to conserve kinetic energy. This process will continue until the point when the parallel velocity is zero. At this point the particles motion up the magnetic field stops. It doesn't have the energy to go further "up" the field. Here it will reverse direction and start falling "down" the field. This you can trap a particle between to "hills" of magnetic field as long as the change in the magnetic field is large enough.

 

[MoreKnowledge]

Hi MoreKnowledge,

The amount of detail in your question is fine.

First the cartoon:
Charged particles gyrate around magnetic fields like beads on a string. Normally they are free to move along (parallel) the magnetic field, but their motion perpendicular to the magnetic field is deflected into circular orbits. Now imagine a loose bundle of finite length stings. Here the beads can easily slide on and off the ends of the stings. You can make it harder for the beads to slide of the ends by squeezing to ends of the strings together so that they're tightly packed. This akin to how the magnetic mirror works.

The actual physics behind the mirror effect is a little more complex. It involves the conservation of energy and the conservation of the magnetic moment. You can show that the quantity $\mu = \frac{m v_\perp}{2B}$ is conserved by a charge particle moving in a slowly varying magnetic field. Here $v_\perp$ is the particles velocity perpendicular to the magnetic field and $\mu$ is called the magnetic moment. Note that $v_\perp$ will increase as the as the magnetic field increases. Now the magnetic field can not do work on a charged particle. This is because the Lorentz force always acts perpendicular to velocity of the particle $\vec F_m = \vec v \times \vec B$ but work is defined as the force in the direction of the velocity $W = \int \vec F_m \cdot \vec v dt = \int \left(\vec v \times \vec B\right) \cdot \vec v dt = 0$. The fact that the work done on the particle is zero implies the the kinetic energy of the particle must be conserved. The total kinetic energy of the particle is the sum of the parallel and perpendicular kinetic energies $KE = \frac{1}{2}mv_\parallel^2 + \frac{1}{2}mv_\perp^2$. To conserver the kinetic energy the square of parallel velocity must decrease as the square perpendicular velocity increases. But the square of the parallel velocity cannot be negative. Thus the conservation of energy places a limit on the maximum allow allow perpendicular velocity.

We now have all the pieces together to understand the magnetic mirror. Let us consider the motion of a charged as it travels into a region of increasing magnetic field.
The particle has some initial perpendicular velocity and parallel velocity. As the particle travels "up" the field the strength of the field increases and the perpendicular velocity increases to conserve of the magnetic moment. As the perpendicular velocity increases the parallel velocity decreases to conserve kinetic energy. This process will continue until the point when the parallel velocity is zero. At this point the particles motion up the magnetic field stops. It doesn't have the energy to go further "up" the field. Here it will reverse direction and start falling "down" the field. This you can trap a particle between to "hills" of magnetic field as long as the change in the magnetic field is large enough.

I hope I'm formatting this correctly. To the earlier response: yes, I did read the Wikipedia article before asking the question. In the Wikipedia article, it illustrates an electron spiraling around what I take to be a magnetic field line.

It makes sense to me that the electron's momentum is conserved while it spirals, but I do not understand why it spirals. A magnetic field in reality is not composed of field lines. They are just a helpful way of representing the extent and intensity of magnetic fields.

So, here the analogy of beads on a string disappears as there are no longer strings. Why do the electrons spiral around, and what do they spiral around? Also, I would like to confirm that I am correctly understanding what you have said. When an electron locks in with a magnetic field, does it travel to the region with the lowest magnetic field strength? Thus, it goes to the valley between the two hills, between the two high intensity regions of the field?

In the Polywell device with its six intersecting magnetic fields, does a trapped electron migrate to the center, the null region? Thank you for all your help!

 

[mfb]

Why do the electrons spiral around

The Lorentz force makes every electron path in a homogeneous magnetic field a helix. You can draw a magnetic field line at the center for visualization, but you don't have to. If that helix leads to a region of increased field strength, the helix windings get more compressed and the velocity component along the magnetic field goes down, until it reaches zero, and reverses. All a result of the Lorentz force.

 

[the_wolfman]

I hope I'm formatting this correctly. It makes sense to me that the electron's momentum is conserved while it spirals

The particles momentum not conserved as it gyrates around the field line. The energy and the magnetic moment are conserved.

As MFB pointed out this is a direct consequence of the Lorentz force. The Lorentz force acts in the direction perpendicular to the particles motion and the magnetic field. If there is a particle moving perpendicular to the magnetic field the Lorentz force when bend the particles trajectory into a circle.

A magnetic field in reality is not composed of field lines. They are just a helpful way of representing the extent and intensity of magnetic fields. So, here the analogy of beads on a string dissapears as there are no longer strings. Why do the electrons spiral around, and what do they spiral around?

Magnetic fields are real. The magnetic field exerts a force on particles that causes that bends their trajectories. The force acts in such a way that the particle motion looks like its gyrating around a field line. While the idea of field lines is a mathematical construct, it is a very useful tool for understanding the 3D structure of vector fields like the magnetic field. In the case of magnetic confinement to lowest order the motion of individual charged particles follows the "hypothetical" field lines.

I would like to confirm that I am correctly understanding what you have said. When an electron locks in with a magnetic field, does it travel to the region with the lowest magnetic field strength? Thus, it goes to the valley between the two hills, between the two high intensity regions of the field?

The particle will bounce back and forth between the two high intensity regions.

 

[MoreKnowledge]

The Lorentz force makes every electron path in a homogeneous magnetic field a helix. You can draw a magnetic field line at the center for visualization, but you don't have to. If that helix leads to a region of increased field strength, the helix windings get more compressed and the velocity component along the magnetic field goes down, until it reaches zero, and reverses. All a result of the Lorentz force.

Does the Lorentz force curve the path of the electron into a circle, but it travels in a helix because the electron is moving from the high strength magnetic field hills down into the valley?

If there is a particle moving perpendicular to the magnetic field the Lorentz force when bend the particles trajectory into a circle.

What happens when the particle is not perpendicular to the magnetic field? Also, does something determine if the particle spirals clockwise or counter-clockwise?

The particle will bounce back and forth between the two high intensity regions.

Will the particle enter the high intensity regions or only bounce between them?
Because the field gradually goes from high to low and back to high intensity, what happens as the particle enters higher and higher intensity regions?Also, have these electron spirals been seen/detected with a probe of some sort, or have they been predicted to exist by calculations?

 

[mfb]

You won't understand magnetic mirrors without the basics. Before you consider magnetic mirrors, please go a step back and consider the motion of electrons in a homogeneous magnetic field. The force is $F=q \vec v \times \vec B$ where q is the charge, v is the velocity vector, B is the magnetic field and $\times$ is the cross product. The force is always orthogonal to both the velocity and the magnetic field, and it depends on the velocity component orthogonal to the magnetic field only.

If you have an electron moving orthogonally to a magnetic field, it will travel in a circle. If the electron also has some velocity component along the magnetic field, this does not influence the force: it will still move in circles but with an additional constant velocity along the direction of the magnetic field. The combined motion is a helix.

What happens when the particle is not perpendicular to the magnetic field? Also, does something determine if the particle spirals clockwise or counter-clockwise?

The direction of the magnetic field.

Every CRT monitor used this concept, and it is easy to visualize the path of electrons in a similar way. Some example images: 1, 2.
On much larger scales, particle accelerators use this concept to keep electrons on circular tracks.

 

[MoreKnowledge]

Sorry to keep bothering you. I just really want to understand.

What happens when the particle is not perpendicular to the magnetic field? Also, does something determine if the particle spirals clockwise or counter-clockwise?

Do I only consider the velocity component of the particle that is perpendicular to the magnetic field? If an electron is traveling diagonally in relation to the magnetic field, do I only consider the electron's velocity perpendicular to the magnetic field? I believe this is what you are saying here:

The force is always orthogonal to both the velocity and the magnetic field, and it depends on the velocity component orthogonal to the magnetic field only.

If you have an electron moving orthogonally to a magnetic field, it will travel in a circle.

It has been said that the electron will travel in a circle due to the Lorentz force. Is the radius of the spiral determined by the velocity of the electron perpendicular to the magnetic field if the magnetic is constant?

 

[mfb]

Do I only consider the velocity component of the particle that is perpendicular to the magnetic field?

Only this component determines the force and therefore the acceleration. The other component is still relevant for the motion of the electron.

It has been said that the electron will travel in a circle due to the Lorentz force. Is the radius of the spiral determined by the velocity of the electron perpendicular to the magnetic field if the magnetic is constant?

Sure. A larger velocity leads to a larger radius.
A larger magnetic field leads to a smaller radius.

 

[MoreKnowledge]

Only this component determines the force and therefore the acceleration. The other component is still relevant for the motion of the electron.
Sure. A larger velocity leads to a larger radius.
A larger magnetic field leads to a smaller radius.

I'm understanding now. Thank you so much! Is there a textbook on this division of physics that you recommend? I've taught myself most of what I know about physics, although what I know may be little compared to you all. It seems to me that I can learn easier through textbooks than from ordinary books. I think that this is because textbooks tend to be better organized.

 

[mfb]

I don't know what you call "ordinary books", but textbooks and university lectures are the usual methods. No idea about good textbooks, sorry. We have a forum discussing various books.

 

[vanhees71]

I'm understanding now. Thank you so much! Is there a textbook on this division of physics that you recommend? I've taught myself most of what I know about physics, although what I know may be little compared to you all. It seems to me that I can learn easier through textbooks than from ordinary books. I think that this is because textbooks tend to be better organized.

I'd start with one of the usual introductory experimental-physics books, e.g., Haliday, Resnick, Walker or Tipler.