On 01/01/2010 03:30 AM, William Beaty wrote: > On Thu, 31 Dec 2009, Harry Veeder wrote: > > billb wrote: >> On Wed, 30 Dec 2009, Craig Haynie wrote: >> > By moving a magnetic field across a conductor, don't we get induction, >> > and hence, electricity? >> >> Not in a toroid inductor with unsaturated core. The ring-shape core >> will shield the inductor against fields coming from nearby magnets. > > For no induction to happen, wouldn't this also require that the magnetic > field (of the permanent magnet) be entirely uniform as experienced by > the torriod? > > Nope. Look at it this way: if b-fields cannot leak out, then there is > zero coupling to external magnets ...and external fields cannot induce > any current in the coil. Or start out by imagining an air-core toroid. > Air-core toroids are self-shielding, and external fields cannot induce > any current in an air-core toroid, even if the fields are nonuniform.
#$%^$%&!! So the problem I had is failure to properly picture a non-uniform magnetic field. First, let me explain that I see why an ideal toroidal coil has no external field -- symmetry and simple arguments regarding the curl of the B field show that it's got to be null outside the torus. In particular, any loop around the outside of the torus must have zero net B field (if we integrate it around the loop), because it encloses a surface which can be entirely outside the torus, which means curl(B)=0 on the whole surface. Second, I can see how conservation of momentum implies an external field can't affect the torus -- interactions must be a two way street, and if the torus can't "talk back" then it can't "listen" either. But third, it *seemed* to me that a change in a non-uniform B field would result in curl(E)=-dB/dt on one side of the torus than the other, and this would result in a net EMF in the torus. The point I missed is that the div(B) and curl(B) must both be zero in free space (absent any electric field) so a non-uniform B field can't just be a bunch of fence posts sticking straight up, with some posts shorter than others. That image has nonzero curl! Rather, the fence posts must bend, and they must bend toward the stronger part of the field. Consequently, the "stronger" part of the field affects less of the torus, the "weaker" part of the field affects more of it (because of the bending) and as the field changes the induced E field in the wires of the torus is balanced out. Gak. I've attached a crude drawing to show sort of what I'm talking about. In the drawing, longer arrows indicate a stronger B field. The direction of the B field relative to the torus, clockwise or counterclockwise, determines which way the induced voltage goes as the field changes strength. As a stronger B field changes it'll induce a stronger E field in the loops of wire in the torus around that portion of the B field. However, note in the drawing that the "weaker" B field, which is going clockwise around the torus, is affecting more of the torus than the "stronger" field, which is going counterclockwise. My explanation is not very clear -- I hope it makes at least a little sense to folks...
<<attachment: B-field-in-torus-2010-01-01.jpg>>
