MyOpenMath/Solutions/Toroid inductance
is a link to a section OpenStax University Physics that discusses the rectangular toroid. This discussion reinforces a number of concepts that are important for scientists and engineers to grasp. One concept is the construction of an integral over a mathematically defined volume. In the textbook the magnetic field was integrated. Here we reinforce that skill by asking students to calculate the volume of a toroid. The serious student should strive to understand every step in this derivation, because methods like this might be required in more advanced courses.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Washer_%28PSF%29.png/220px-Washer_%28PSF%29.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Washer_4_calculus_integration.svg/220px-Washer_4_calculus_integration.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Differential_change_in_area_and_radius_of_circle.svg/220px-Differential_change_in_area_and_radius_of_circle.svg.png)
The toroidal inductor gives us the opportunity to:
- Learn how calculus can be used to find the volume and area of objects.
- Use Amphere's law to find the magnetic field for certain geometries with high symmetry.
- Calculate the self inductance of certain coils.
Volume
[edit | edit source]This is a good example of how a double integral can be used to find the volume of shapes with a high degree of symmetry. Let denote a volume and denote an area, with the understanding that denote a volume and are differentials that can be integrated as the limit of Reimmann sums. The figure to the right resembles that of a rectangular washer that had its top surface shaved down. The rectangular area shaded in yellow is an area differential:
where is a function of . For example, the function might be where and are constants. This function is defined with a range defined by two other constants :
The figure shows two identical cross-sectional areas that can be integrated as, , but this step would be premature. Instead we sweep the rectangle of differential (small) area around the circumference to obtain the differential volume:
Note how this technique offers a simple proof for the area of a circle, assuming that the circumference is known to be 2πr.