Section 4.2 · Home exercises · 3 of 4
Home exercise 4.2.3: Energy stored in a toroid
Energy stored in a toroid
#A toroid of closely wound turns has a mean radius and a cross-section of area much smaller than , so that may be taken as uniform over the cross-section. Its core has permeability and the winding carries a current . Find inside the core from Ampère's law, the self inductance of the toroid, and the stored magnetostatic energy , once from and once from . Check that the two agree.
A toroid is a solenoid bent into a ring: Ampère's law on the mean circle links all turns. The field is confined to the core, so the energy integral runs over the core volume only.
By symmetry inside the core. Ampère's law on the circle of radius links all turns:
since a circle outside the core links either no turns or turns in and turns out.
Each turn links :
From the circuit form,
From the field: the core volume is and is uniform in it,
the same.
Compare with the field integral in the results.
The two routes agree because all the field is inside the core: over all space is for a single circuit. With the numbers of Worked example 4.2.1 (, , , , ): , and . The toroid is the solenoid of Problem 4.2.3 with turns per metre , and Eq. (4.2.21) gives the same .