View source for MyOpenMath/Solutions/Toroid inductance

{{:User:Guy vandegrift/T/Title}}
https://openstax.org/books/university-physics-volume-2/pages/14-2-self-inductance-and-inductors#CNX_UPhysics_31_02_Toroid

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.

[[File:Washer (PSF).png|thumb|The technical term for a this "washer" shape is rectangular [[w:toroid|toroid]].]]
[[File:Washer 4 calculus integration.svg|thumb|This is not a rectangular toroid because the top surface is curved.]]
[[File:Differential change in area and radius of circle.svg|thumb|The yellow shaded portion resembles a rectangle of width dr.  The length is somewhere between 2&pi;r and 2&pi;(r+dr).  In most cases the area can be taken to be 2&pi;dr in the limit that dr goes to zero.  Integration of 2&pi;(r+dr) from r=0 to r=R yields the area of a circle to be &pi;R<sup>2</sup>.]]

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==
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 <math>V</math> denote a volume and <math>A</math> denote an area, with the understanding that <math>dV</math> denote a volume and <math>dA</math> are differentials that can be integrated as the limit of [[w:special:permalink/919572791|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:

:<math>dA=h\,dr</math>

where <math>h=h(r)</math> is a function of <math>r</math>.  For example, the function might be <math>h(r)=Cr^{-\alpha},</math> where <math>C</math> and <math>\alpha</math> are constants.  This function is defined with a range defined by two other constants <math>(r_1,r_2)</math>:

:<math>r_1<r<r_2</math>

The figure shows two identical cross-sectional areas that can be integrated as, <math>\int_{r_1}^{r_2}Cr^{-\alpha}\,dr</math>, but this step would be premature. Instead we sweep the rectangle of differential (small) area <math>h\,dr</math> around the circumference <math>2\pi</math> to obtain the differential volume:

:<math>dV=2\pi r\,dA = 2\pi r\,h(r)dr</math>

:<math>V=2\pi C\int_{r_1}^{r_2}r^{-\alpha+1}dr</math>

Note how this technique offers a simple proof for the area of a circle, assuming that the circumference is known to be 2&pi;r.

==Under construction==
===Finding the magnetic field===
===Finding the self inductance===
===Understanding magnetic energy===

[[Category:MyOpenMath]]