PHYS208 Fundamentals of Physics II

Application of the Binomial Theorem

For situations involving distribution of a net charge over an extended region, the calculated electric field dependence may be checked in the limit where the point of evaluation is far from the charge distribution. When a finite amount of charge is the source, the far-field behavior of the electric field should behave as a point charge of that amount. Often the binomial theorem is useful in considering that limit:

The binomial theorem is especially useful in converting negative or fractional exponents into ordinary polynomial expressions from which the leading-order dependence may be determined. For example, for small x the binomial theorem can be used on the following fractions to determine the linear dependence on x:

Consider as an example an expression of relevance to a recent takehome quiz. The leading-order dependence of the following expression involving r and a may be determined for the case of a << r, i.e. far away:

Derivation of the Binomial Theorem

"http://www.physics.udel.edu/~watson/phys208/binomial-example.html"
Last updated March 10, 1998.
Copyright George Watson, Univ. of Delaware, 1998.