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:
Last updated March 10, 1998.
Copyright George Watson, Univ. of Delaware, 1998.