## 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:

"http://www.physics.udel.edu/~watson/phys208/binomial-example.html"

Last updated March 10, 1998.

Copyright George Watson, Univ. of Delaware, 1998.