Magnetic Field inside a Toroid Task number: 1784 Derive the formula for the magnitude of the magnetic field inside a coil that has the shape of a torus whose minor radius is much smaller than the lenght of the central circle.
A toroid using a square.
A torus is a type of toroid.
In mathematics, a toroid is a surface of revolution with a hole in the middle, like a doughnut, forming a solid body. The axis of revolution passes through the hole and so does not intersect the surface.[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topologicalgenus, g, of 1 or greater. The Euler characteristic Ï of a g holed toroid is 2(1-g).[2]
A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):
- V=2ÏRA{displaystyle V=2pi RA}
- S=2ÏRC{displaystyle S=2pi RC}
Which gives for square and circular sections:
The volume and surface of a toroid with square section of side a are given by
![Toroid Toroid](http://patentimages.storage.googleapis.com/US20120235528A1/US20120235528A1-20120920-D00000.png)
- V=a22ÏR{displaystyle V=a^{2}2pi R}
- S=8aÏR{displaystyle S=8api R}
The volume and surface of a toroid with circular section of radius r (torus) are given by
- V=2r2Ï2R{displaystyle V=2r^{2}pi ^{2}R}
- S=4rÏ2R{displaystyle S=4rpi ^{2}R}
See also[edit]
Notes[edit]
- ^Weisstein, Eric W.'Toroid'. MathWorld.
- ^Stewart, B.; 'Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces', 2nd Edition, Stewart (1980).
External links[edit]
- The dictionary definition of toroid at Wiktionary
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