Magnetic dipole moment of toroid

Question: Must every magnetic configuration have a north and south pole?

Answer: Not necessarily. True only if the source of the field has a net non-zero magnetic moment. This is not so for a toroid or even for a straight infinite conductor.

(source: uni-wuppertal.de)

I understand that a straight infinite conductor will not experience a torque in a uniform magnetic field. It will only experience a net force based on its orientation. And so it doesn't have a net magnetic moment.

But I'm unable to visualize how a toroid will behave in a uniform external magnetic field. Since the answer to the above question says that a toroid doesn't have a magnetic dipole moment, I can conclude that it doesn't experience a torque. But does it experience a net force? Please explain how a toroid behaves in an external magnetic field at the basic level of each loop of wire.

• First, all known fundamental sources of magnetism are, at first approximation, magnetic dipoles. There are no observation of free magnetic monopoles yet. Second, an infinite wire may not have, strictly speaking, a net magnetic moment, but how is current supposed to flow in an open loop. It is effectively unphysical on that respect. Third, it is unclear what you ask... What is the toroid exactly? Is there a current? Commented Dec 20, 2016 at 11:43
• @G. Bergeron 1. All sources of magnetism are either magnetic dipoles or magnetic configurations without any poles. The toroid and straight infinite current-carrying conductors are examples of the later. 2. Current can flow through a straight wire which has +ve and -ve charge reservoirs at either end which aren't connected to each other. Isn't that physically possible?(but I get that it can't be infinite). 3. Thank you for notifying me. I've updated the question. Commented Dec 21, 2016 at 7:53
• First of all, it is completely untrue that all sources of magnetism either come as dipoles or no poles at all (unless the field is zero). The meaning of poles takes its origin in the multipolar expansion and thus requires that the field be asymptotically zero, which is not the case for objects of infinite extent. A naive calculation for vector potential with objects of infinite extension leads to absurd results that depends on the topology of the space. Commented Dec 21, 2016 at 9:20
• The case of the charge reservoirs leads to what is essentially a dipole antenna and while it's true it may not have a dipole moment, it will have non-zero higher multipole moments. But now, there will be a net torque from an external magnetic field. For the case of a perfect toroid, even though the external magnetic field is vanishing, there will be multipolar moments of all orders. See en.wikipedia.org/wiki/Toroidal_inductors_and_transformers Commented Dec 21, 2016 at 9:29

1 Answer

A torus can be a magnetizable object; what you describe, is a solenoid-like winding bent into a toroidal shape. So, the current is circulating through the hole in the torus, on a minor-diameter route. Such current is called 'poloidal'.toroidal and poloidal

That current is a self-shielding magnetic geometry (and toroids with windings to create poloidal current are highly prized as inductors, because they don't accept external interference, nor radiate). And, yes, the fact that no external field is coupled means that there's no net torque or force.

There is another useful property, in that a poloidal wound magnetizable torus can be driven nonlinear due to added external field which saturates the magnetizable material. Such a saturation imbalances the flux in the torus, and creates coupling (which can be used to detect the external field). This is the principle on which flux-gate magnetic field detection is based.

• Para 1: Thank you for clarifying what a toroid is. Para 2: I understand why a toroid doesn't radiate a magnetic field from Ampere's Circuital Law. But please explain why and how the toroid doesn't accept external interference. Also please explain what coupling of external field means. Para 3: Interesting idea. Can you suggest sources for me to read up for getting better insight into what you have said. P.S. I'm in high school. Sorry if my questions seem naive. :) Commented Dec 21, 2016 at 8:03
• Okay. A poloidal current around a toroidal form has no monopole, dipole, or other external B field. Because of this, by Newton's third law, it cannot exert magnetic force on an external magnet (because the 'equal, opposite reaction force on the external magnet cannot come from a magnetic field of the toroid). Commented Dec 21, 2016 at 9:12
• As for external interference, the connection between magnetism of a poloidal current (I have to not say 'toroidal current' because that means a circular current that DOES create external magnetic field) is broken; just as it does not exert force outside the torus, it cannot have the 'equal and opposite' reaction to a magnetic force from outside. I'm not sure how to get better insight into this; it took me a few years of college and grad school. Commented Dec 21, 2016 at 9:18
• The illustration shows a winding that in addition to going poloidally, makes one turn (well, 95% of one turn) in the toroidal direction at the same time. That means it is not a purely noninterecting winding. Commented Dec 21, 2016 at 9:24