Toroid

Toroids are essentially the shape of donuts/bagels. Devices in this shape show up occasionally in electrical engineering, like in some inductors or transformers.

Electromagnetism

Remember a circle indicates the field points out of the page, while an X indicates the field points into the page. Look carefully at the question before drawing the X and dots. The X usually is on the outside but can be the other way around.

We take a few central simplifications: that the wire is replaced by a series of rings adjacent to each other with each ring having a current flux flowing in it, $I={\iint }_S\vec{J}\cdot d\vec{S}$. We can use cylindrical/polar symmetry where the field only depends on $r$. We’ll have two directions of the field, one for inside and the other for outside (where outside is negative of inside).

$$\vec{F}=\pm f(r)\hat{\theta }$$

Two general cases: one is where the Amperian loop is outside the toroid, and one is where its inside. In the case where its outside, the currents enclosed cancel out (from outer and inner parts) and equal 0, so the field equals 0. In the case where its inside the toroid:

• LHS of Stokes’ theorem is $2\pi rf(r)$.
• RHS is more difficult to solve.
  • If we have number of turns, we have $n\cdot i_{enc}$.

This overall results in a field:

$$\vec{B}(\vec{r})=\{\begin{array}{ll}\frac{{\mu }_{0}Ni}{2\pi r}\hat{\phi }& \text{inside the coil}\\ \vec{0}& \text{outside the coil}\end{array}$$

2021 final

Here’s a past final exam problem:

See also

• Solenoid, which shows up in a similar context