SMALL MATHEMATICS


Tim Healy, Santa Clara University


"Small Mathematics" is a collection of well-known approximations which are often of use in electomagnetics. In the following expressions the letter "s" is used to mean a number which is small compared with one. The approximations include:

cos(s) ~ 1

sin(s) ~ s

This says that the cosine of a small number is approximately equal to one, and the sine of a small number is approximately equal to the small number. Of course we have not addressed here the always important question of whether in any particular application it is wise or desirable to use the approximation. Some additional approximations follow.

tans~s

ln(1+s)~s

1/(1-s)~1+s

sqrt(1+s)~1+s/2

nthrt(1-s)~1-ns

Students might find it interesting to check the accuracy of these approximations by using a hand calculator and some examples.


EXAMPLES OF APPLICATIONS OF SMALL MATHEMATICS

Example 1: Electric Field Due to a Circular Disk of Charge

The electric field directly above a round disk with a charge surface density of rhos is (See, for example, Sadiku, 'Elements of Electromagnetics, Second Edition', Saunders, 1994) given by:

E = (rhos/2e)(1-h/(sgrt(hsq - asq))

where e is the permittivity, h is the distance directly above the disk, a is the radius of the disk, hsq means h squared, and asq means a squared. (The reader may wish to refer to Sadiku, or a similar text, and/or sketch the system and write out this equation before proceeding).

Let us assume first that a>>h, that is, the point of measurement is 'very close' to the surface of the disk. Then the mathematical expression reduces, without the use of Small Mathematics, to the approximation:

E~rhos/2e.

which is well known to be the electric field over an infinite sheet of charge. That is, the approximation shows that when we are very near the surface (h is small compared to a), the disk looks like an infinite surface.

If, on the other hand, h>>a, that is, the point is far away from the disk, the application of small mathematics to the square root yields the approximation:

E~rhos(pi)asq/4(pi)ehsq

The numerator is the charge density multiplied by the surface area, which is the total charge on the disk. And hence the approximation is seen to be the charge divided by 4 times pi times the permittivity times the square of the distance from the disk. But this is the well known expression for the electric field due to a point charge. Hence, we see that the disk acts like a point charge if we are a great distance from it.


Example 2: The Capacitance of a Coaxial Capacitance

The capacitance of a coaxial capacitor is known to be:

C = 2(pi)eL/ln(b/a)

where (pi) is 3.14159..., e is the permittivity, L is the length of the capacitor, ln(.) is the natural log, b is the outer radius and a is the inner radius of the coax structure.

Let us rewrite the denominator as:

ln(b/a) = ln(1+ (b-a)/a)

Next let us assume that a and b are almost equal, that is, the cylindrical plates are very close to each other. Then (b-a)/a is small, and we can apply one of our Small Mathematics approximations to yield:

C~2(pi)aeL/(b-a)

The quantity 2(pi)aL is the surface area of the inner cylinder, and is approximately equal to the surface area of the outer cylinder. And (b-a) is the distance between the two cylinders. Hence the coaxial capacitance is seen to approximate the capacitance of a parallel plate capacitor

C = eA/d

as we would expect.


Persons interested in commenting on this page, or suggesting additions or changes are invited to contact: thealy@scuacc.scu.edu

I would particularly appreciate any thoughts on how to write better equations in HTML. I could of course use graphics and a gif file, but I hate to waste the memory.