# How does a magnetic bottle work?

• yoran
But isn't it so that the direction of the electric field is opposite the change of flux in the magnetic field? If you have a weaker magnetic field in the center and a strong one at the ends, then if you move from the center to an end, the magnetic flux will be increasing. Therefore, I assume (law of Faraday) that the electric field set up by the changing magnetic field will be directed opposite the change of magnetic field, i.e the electric field will point towards the center. Because \overline{F_e} = q\overline{E}, if q is negative then the charge will be ejected out of the bottle and if q is positive, the charge will experience a force towards the center so will

#### yoran

Hi,

I'm wondering why charged particles stay trapped in a magnetic. Assume the magnetic bottle is oriented in the x-direction. Then a particle will keep oscillating in the x-direction while making circle movements in the y- and z-direction.
But why does it oscillate in the x-direction? Since the magnetic force does no work on a particle because it is always directed perpendicular to the velocity, the kinetic force of the particle stays constant. When oscillating, there must be some point where the speed in the x-direction is zero, which means that the kinetic energy must change. How is that possible? I suppose the magnetic force alone can't be responsible for the oscillation?

Thank you.

The magnetic bottle is made in such a way that there is a significantly stronger magnetic field at each end than at the center. The particle sees a changing magnetic field, which by maxwell's equations will accelerate the particle - via an electric field intermediate. The total kinetic energy of the particle does not change; the x kinetic energy gets converted to y and z KE; after it changes direction this process reverses etc.
Does that help?

lzkelley said:
The magnetic bottle is made in such a way that there is a significantly stronger magnetic field at each end than at the center. The particle sees a changing magnetic field, which by maxwell's equations will accelerate the particle - via an electric field intermediate.

i think that will happen with a constant magnetic field and moving charged particles. they'll go in circles.

absolutely; the effect happens with uniform B fields as-well. There you don't get the mirror effect however.

Ok thank you. I guess it will only work with positively charged particles because imagine a particle being in the center of the bottle and moving to the left end. The magnetic field gets stronger and stronger so the magnetic flux is increasing. Therefore, the change in flux is positive. According to Faraday's law of induction, an electric field is set up that is opposite the change in flux, so opposite the magnetic field. Thus the electric field will point to the center of the bottle. Therefore, I guess that only a positively charged particle will oscillate as a negatively charged particle will be accelerated out of the bottle. Am I correct (you can trap negative charges by putting a stronger field in the center of the bottle and a weaker one at the ends)?

yoran said:
Ok thank you. I guess it will only work with positively charged particles because imagine a particle being in the center of the bottle and moving to the left end. The magnetic field gets stronger and stronger so the magnetic flux is increasing. Therefore, the change in flux is positive. According to Faraday's law of induction, an electric field is set up that is opposite the change in flux, so opposite the magnetic field. Thus the electric field will point to the center of the bottle. Therefore, I guess that only a positively charged particle will oscillate as a negatively charged particle will be accelerated out of the bottle. Am I correct (you can trap negative charges by putting a stronger field in the center of the bottle and a weaker one at the ends)?

there's no qualitative difference between positive charge and negative charge. which one that was assigned "+" is a matter of convention. now negatively charged particles are usually electrons and positive charged particles are usually an atom with an electron missing. the charge to mass ration of a sole electron will be much greater, in magnitude, than the charge to mass ratio of an atom missing an electron. so the heavier charged particles might not have as tight of a circular loop (in a magnetic field) as the lighter charged partilces. but their masses are identical (and they have charge that is negative of each other), then the only difference between the two (positive vs. negaitive charged) is that one will spin around the mag field one way and the other will spin around the opposite manner.

But isn't it so that the direction of the electric field is opposite the change of flux in the magnetic field? If you have a weaker magnetic field in the center and a strong one at the ends, then if you move from the center to an end, the magnetic flux will be increasing. Therefore, I assume (law of Faraday) that the electric field set up by the changing magnetic field will be directed opposite the change of magnetic field, i.e the electric field will point towards the center. Because $$\overline{F_e} = q\overline{E}$$, if q is negative then the charge will be ejected out of the bottle and if q is positive, the charge will experience a force towards the center so will do an oscillation. Is it correct what I'm saying?

You're on the right track - and you're using some really good arguments, but ultimately you're wrong, here's why: the "drift" caused by the increasing strength of the magnetic field is called the "grad-B drift" because it involves the Gradient of the magnetic field (B), I'm not sure if you're familiar with the gradient, but its a device that represents the change in something when its changing in more than just one direction (where we simply use the derivative).

Anyway! - the particle experiences a force proportional to the cross product between the gradient of B, and the B field itself, and is NOT DEPENDENT ON THE CHARGE OF THE PARTICLE ---> F = B x Grad(B) ---> which means that the force will be perpendicular to both the magnetic field lines, and the direction in which they are increasing in strength (towards the ends of the bottle). The force will be in the directed inwards (relative to the bottle) and a little bit in the plane of normal gyration, independent of what the charge is.

Allright thank you. This force you are speaking of, is it a result of Maxwell's equations? Because I never read about this force...

lzkelley said:
You're on the right track - and you're using some really good arguments, but ultimately you're wrong, here's why: the "drift" caused by the increasing strength of the magnetic field is called the "grad-B drift" because it involves the Gradient of the magnetic field (B), I'm not sure if you're familiar with the gradient, but its a device that represents the change in something when its changing in more than just one direction (where we simply use the derivative).

Anyway! - the particle experiences a force proportional to the cross product between the gradient of B, and the B field itself, and is NOT DEPENDENT ON THE CHARGE OF THE PARTICLE ---> F = B x Grad(B) ---> which means that the force will be perpendicular to both the magnetic field lines, and the direction in which they are increasing in strength (towards the ends of the bottle). The force will be in the directed inwards (relative to the bottle) and a little bit in the plane of normal gyration, independent of what the charge is.

Hmm...~ lzkelley, I am not sure what you mean by gradient of the magnetic flux "vector" field. Last I checked, gradient cannot be applied onto a "vector" field function, but can be applied onto a "scalar" function. Do you mean by divergence of the magnetic flux "vector" field? If so then, By divergence theorem on magnetic flux vector field, due to the fact that magnetic monopole has yet to be discovered, the divergence of any naturally occurring phenomena currently known to man kind of magnetic flux vector field function yields the number zero. I am kind of confused following your kind response and the force equation you have kindly listed above: F = B x Grad(B). Please enlighten me, I like to understand things in various ways and look at things from different angles. Thank you :).

I will share my understanding to this question. The physical reasoning of a magnetic bottle mainly takes magnetic field line (check out this http://rt210.sl.psu.edu/phys_anim/EM/magnetic_bottle2_thm.gif" [Broken]) and Lorentz Force Law. more specifically just the magnetic force portion: vector_F = q*cross(vector_v,vector_B); or F = q*vxB. If you look at the picture I linked above which provides magnetic field lines, you should see that the magnetic field spreads out toward middle region(This means a decrease in magnetic flux intensity). This means in addition to the axial field component there is also radial component(Try hooking up two coils and pump some current in the same direction and same winding then use a compass to check field directions if you find my described situation unconvincing). The key to the charge accelerating in the axial direction depends on the radial component(Try crossing the velocity of a circular path charge with a radial field component then you will see a force in the axial direction). I think upto this point Yoran's question is partially answered. That is the crux in the charge accelerating in the axial direction I think, which is quite subtle to me at least when I first saw this.

Yoran said:
Hi,

I'm wondering why charged particles stay trapped in a magnetic. Assume the magnetic bottle is oriented in the x-direction. Then a particle will keep oscillating in the x-direction while making circle movements in the y- and z-direction.
But why does it oscillate in the x-direction? Since the magnetic force does no work on a particle because it is always directed perpendicular to the velocity, the kinetic force of the particle stays constant. When oscillating, there must be some point where the speed in the x-direction is zero, which means that the kinetic energy must change. How is that possible? I suppose the magnetic force alone can't be responsible for the oscillation?

Thank you.

Does kinetic energy change? The answer is no, it does not. Is magnetic force responsible completely for the oscillation? Yes, it does. Having hopefully convinced you the existence of a radial field component, try crossing the radial magnetic flux component with the charge with axial velocity should yield a force decelerating the charge in the circular direction, or to be more technically correct, a force pointing in the direction tangential but negative in magnitude to the circular path component of the charge, which implies the charge slows down its circular motion. That means even though the charge gains axial component of the velocity, the circular component(alright this isn't a good technical description of the velocity in THAT direction but you get my point) slows down, if you actually perform the calculation this yields a net change of 0 Joules in energy. Therefore, as Izkelley has generiously pointed out in his/her post near the beginning of this entire conversation, no work has actually been done by the magnetic field, which obeys the observation given in various textbooks. Hope that I am not too far from being correct, and if I just so happen to be correct, then I hope I've made it clear and my point across to you my comrades in Physics~ :)

Edit 1:
Also if you understand what I am referring to by the circular path thingy in the last paragraph. Please kindly let me know how I can phrase it better because my language has failed me miserably in this case... sadly enough T_T~

Edit 2:

Nevermind, I think I'll just call it the phi component~

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Think of a weak focusing cyclotron with a vertical Bz field that is stronger near the pole tips than it is at the central orbit. This is a magnetic bottle. The field lines bulge out radially a little bit at the central orbit. There is no azimuthal magnetic field, only a Bz field and a Br field. This gives rise to vertical focusing that keeps the protons from hitting the pole tips. When there is no azimuthal RF field, the protons keep circulating at constant energy (no acceleration or deceleration). So the only requirements are: No electric field, Btheta = 0, a strong Bz, and by necessity (meaning curl B = 0) a weak radial magnetic field.

. The magnetic field is constant (independent of time). This is not like the betatron, that uses dB/dt inside the orbit to accelerate electrons.

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Here are annimations from Penn State on the motion of a charged particle in a magnetic field (go down about half way):
http://rt210.sl.psu.edu/phys_anim/EM/indexer_EM.html [Broken]
Here are two annimations of a charged particle in a uniform solenoidal field, one on a charged particle in a toroid, and then two on a charged particle in a magnetic bottle. The stronger fields on each end act as mirrors so charged particles within certain momentum ranges are captured. There is complete axial symmetry, so only the radial and axial magnetic fields are functions of r and z.
 For those who want to pursue the equations of motion in a magnetic bottle, here is a complete derivation of charged particle motion in a cyclotron magnetic field, which is a form of magnetic bottle:
http://www.physics.rutgers.edu/cyclotron/papers/12_inch_magnet_studies_4.pdf

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