Section 5.7 Multipole Expansions (Power vs. Laurent Series)
Whether or not something is large or small must ALWAYS be constrasted to something else.
In many physical situations, an experimenter (observer) is situated far from the objects being observed. Think about an experimenter observing atoms, where the experimental apparatus in the laboratory is far from the atoms compared to the size of the atoms being observed. Or think about an astrophysicist who is necessarily very far from the galaxy they are observing compared to the size of the galaxy. It doesn’t matter about the size of the object being observed, even in comparison to the size of the observation apparatus. What matters is the size of the object being observed compared to the distance from the measuring apparatus.
In these cases, an important approximation scheme is a multipole (or Laurent series) expansion, rather than a power series expansion.
For example, in Activity 5.5.1, we found the two series expansions
\begin{align*}
V(0,0,z) \amp =
{2Q\over 4\pi\epsilon_0}\, {1\over D}
\left( 1 + {z^2\over D^2} + {z^4\over D^4} + ...\right)
\amp\text{ for} |z|\lt D\\
V(0,0,z) \amp =
{2Q\over 4\pi\epsilon_0}\, {1\over |z|}
\left( 1 + {D^2\over z^2} + {D^4\over z^4} + ...\right)
\amp\text{ for} |z|>D
\end{align*}
for the electrostatic potential due to two point charges \(+Q\) and \(+Q\) that are placed on the \(z\)-axis at \(z=D\) and \(z=-D\text{,}\) respectively.
To obtain these expressions, we used the binomial series from Section 4.7 which is only valid when its dimensionless parameter is less than 1. Since \(D\) is the position of the (fixed) charges and \(z\) is the (variable) distance to the measuring apparatus, the first series is valid between the two charges, for \(z\lt D\text{,}\) and the second series is valid far from the charges, for \(D\lt z\text{.}\)
If you are thinking of the symbol \(z\) as a variable, then technically, only the first series is a power series. The variable \(z\) is raised to positive powers. Since the second series has some negative powers of the variable \(z\text{,}\) mathematicians would call it a Laurent series. Since it has only negative powers and is therefore valid far from the charged source, physicists would call it a multipole expansion.
Multipole expansions reflect the progress of science. The first term of the expansion, which varies like \({1\over z}\text{,}\) tells us what we can measure when our measuring apparatus is relatively crude. Subsequent terms, which are successively smaller, vary like \({1\over z^2}\text{,}\) \({1\over z^3}\text{,}\) etc.. They become relevant as our measuring apparatus improves over time. (In our example, the \({1\over z^2}\) happens to be zero.)
The terms in a multipole expansion are important because they tell us how the source is distributed in space. In our example of charges, we recognize the first term \({2Q\over 4\pi\epsilon_0}\, {1\over |z|}\) as just the potential due to a point charge. We can use a measurement of this term, called the monopole term, to tell us the total charge, in this case \(2Q\text{.}\) The second term is called the dipole term; it tells us how the center of positive charge is separated from the center of negative charges. It would be the dominant term (the first non-zero term) if the total charge were zero. For a perfect dipole, it would be the only term. The third term is called the quadrupole term. It would be the dominant term if the charge distribution, for example, were two dipoles, oriented oppositely.
