4 ms·

> It's impossible to make a magnet in the shape of a ball that squishes on all sides equally... Ah, my favorite theorem, The Hairy Ball Theorem! https://en.m.

> It's impossible to make a magnet in the shape of a ball that squishes on all sides equally... Ah, my favorite theorem, The Hairy Ball Theorem! https://en.m.wikipedia.org/wiki/Hairy_ball_theorem https://en.m.wikipedia.org/wiki/Hairy_ball_theorem

vecter 10y ago
Can you explain how the Hairy Ball Theorem applies to the spherical magnetic field? My first intuition is that the magnetic field would not be a tangent field because you'd want all of the force lines to point inwards to the center of the sphere. However, based on what you said, it sounds like my intuition is wrong. To phrase it another way, could you explain why the magnetic field is tangent to the sphere and not normal to it?
- evanb 10y ago
  There are two things here. The force magnetic fields apply is always perpendicular to the field: F ~ v × B (right hand rule, and all that). So if you want F to point inward, the one component of B that doesn't matter is its radial component---ie. the tangent component is all that counts. But it's not clear to me that you can't compensate for the restrictions of the hairy ball theorem by having a nonuniform velocity distribution. Of course, there's no "hairy torus" theorem (because it's not true!) and this immediately suggests the tokamak design. It's also not possible to take a sphere and have the radial magnetic field point inward everywhere (or outward everywhere), because Maxwell's equations prohibit magnetic monopoles.
  - vecter 10y ago
    It's been too long since E&M. I forgot about magnetic fields being normal doh. And of course, the last point about no magnetic monopoles should be sufficient to make a spherical field impossible.
- CamperBob2 10y ago
  I'd guess that it has something to do with eddy currents induced within the contained plasma. The HBT probably implies that these can't be perfectly symmetrical, and if they aren't symmetrical the whole thing will fall apart.
- wtallis 10y ago
  Charged particles will spiral along magnetic field lines. Thus, you want to have the field lines only be parallel to the confinement surface, so that the particles will migrate around the surface while doing their smaller-scale spiraling, instead of migrating outwards along field lines that exit the confinement.

vecter 10y ago

Can you explain how the Hairy Ball Theorem applies to the spherical magnetic field? My first intuition is that the magnetic field would not be a tangent field because you'd want all of the force lines to point inwards to the center of the sphere. However, based on what you said, it sounds like my intuition is wrong. To phrase it another way, could you explain why the magnetic field is tangent to the sphere and not normal to it?

evanb 10y ago
There are two things here. The force magnetic fields apply is always perpendicular to the field: F ~ v × B (right hand rule, and all that). So if you want F to point inward, the one component of B that doesn't matter is its radial component---ie. the tangent component is all that counts. But it's not clear to me that you can't compensate for the restrictions of the hairy ball theorem by having a nonuniform velocity distribution. Of course, there's no "hairy torus" theorem (because it's not true!) and this immediately suggests the tokamak design. It's also not possible to take a sphere and have the radial magnetic field point inward everywhere (or outward everywhere), because Maxwell's equations prohibit magnetic monopoles.
- vecter 10y ago
  It's been too long since E&M. I forgot about magnetic fields being normal doh. And of course, the last point about no magnetic monopoles should be sufficient to make a spherical field impossible.
CamperBob2 10y ago
I'd guess that it has something to do with eddy currents induced within the contained plasma. The HBT probably implies that these can't be perfectly symmetrical, and if they aren't symmetrical the whole thing will fall apart.
wtallis 10y ago
Charged particles will spiral along magnetic field lines. Thus, you want to have the field lines only be parallel to the confinement surface, so that the particles will migrate around the surface while doing their smaller-scale spiraling, instead of migrating outwards along field lines that exit the confinement.