Example Problem - Toroid

Problem: Concerning Ampere's law Let look at a coil around a toroid shown in the figure below. The coil has N = 20 turns around the toroid. The toroid has an inside diameter of ID = 4 cm and an outside diameter OD = 5 cm. Determine the field intensity H along the mean path length within the toroid with a current i = 2.5 A.

Figure created by Kirk Betz

Solution:

Do symmetry the magnetic field intensity Hm along a circular contour within the toroid is constant. We can find the mean radius by

$$\begin{gathered} r_m=(\frac{1}{2})\frac{OD+ID}{2}=2.25 \\ cm \end{gathered}$$

Using the mean radius the mean path of length $l_m$ can be calculated.

$l_m=2\pi r_m=0.141$

With Ampere's Law (below) the field intensity along the mean path can be Found.

$H_m=(\frac{Ni}{l_m})$

Finally teh H_m can be calculated.

$$\begin{gathered} H_m=\frac{20x2.5}{.141}=354.6 \\ A/m) \end{gathered}$$

Since the width of the toroid is much smaller than the mean radius $r_m$ we can assume a uniform $H_m$ throughout teh cross-section of the toroid.