# Why is the magnetic field outside a toroid 0?

Finding the magnetic field for a straight wire using the right hand rule shows that the field is around the wire.

Applying this same concept for a toroid, the magnetic field inside is central to the loop of wires so that makes sense.

However, isn't there also a magnetic field outside in a direction perpendicular to each point of the wire, spiraling the toroid?

In a loop of wire, there is a central magnetic field, but isn't there also some magnetic field outside of it?

Place your circular Amperian loop outside the toroid.

By definition, Ampere's Law states that integration of magnetic field over an amperian loop is equivalent to constant μ multiplied by current enclosed.

Now, look from the top view. You will observe that current that flows at outer circumference and inner circumference of toroid are in opposite direction to each other. Therefore, current enclosed by Amperian loop is zero. Thus, magnetic field outside the toroid is zero.

jtbell
Mentor
In a loop of wire, there is a central magnetic field, but isn't there also some magnetic field outside of it?

For a single loop of wire, yes. For a lot of loops arranged as a toroid, their contributions cancel outside the toroid, giving the result that is most easily calculated using Ampere's Law. In principle one can also show it by integrating the contributions from each small segment of wire in each loop, using the Biot-Savart law. However, I've never seen it done this way. Too complicated. I think it would have to be done numerically rather than analytically.

Good question.

The current could be constant, or changing in time, such as sinusoidal AC current resulting in a changing electric fields.

Ideally, the current is uniform sheet of current circulating through the throat of the torus without a component of the current circulating around the circumference.

In a degenerate case, a single circular loop of wire is also a single wind around a torus, oriented somewhat askew, so that what can be said about a single circular loop may also apply to the toroid. In this case, the external magnetic field is nonzero, and this applies to the torus as well. This is an example of nonuniform flow on a torus.

In non-ideal conditions the windings are not uniform, and the permeability of the core material is not infinite. In practical terms these are "toroidal core inductors."

tech99
Gold Member
Good question.

The current could be constant, or changing in time, such as sinusoidal AC current resulting in a changing electric fields.

Ideally, the current is uniform sheet of current circulating through the throat of the torus without a component of the current circulating around the circumference.

In a degenerate case, a single circular loop of wire is also a single wind around a torus, oriented somewhat askew, so that what can be said about a single circular loop may also apply to the toroid. In this case, the external magnetic field is nonzero, and this applies to the torus as well. This is an example of nonuniform flow on a torus.

In non-ideal conditions the windings are not uniform, and the permeability of the core material is not infinite. In practical terms these are "toroidal core inductors."
It has been found in practice that a toroid does not provide complete containment of the magnetic field because there is something called the "one turn effect", where the spiral winding has a component acting around the ring.