In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel functions. His expansion is somewhat confusing, and I was wondering any other reference doing the same expansion, in some other manner?

  • $\begingroup$ what other resourcces have you already looked at? $\endgroup$ – Sanya Jun 30 '16 at 16:19
  • $\begingroup$ @Sanya I tried looking at Zangwill's Modern Electrodynamics, he doesn't develop this there. Also, Google yielded nothing. $\endgroup$ – JonTrav1 Jun 30 '16 at 16:41
  • $\begingroup$ en.wikipedia.org/wiki/… for the spherical harmonics; and I am not too sure anymore, but Landau & Lifshitz should treat the problem of multipole expansion at least in spherical harmonics. As soon as the principle is clear, maybe Jackson becomes more approachable. $\endgroup$ – Sanya Jun 30 '16 at 17:43
  • $\begingroup$ Are you comfortable with the addition of angular momentum of quantum mechanics? Then another way to develop the multipole expansion is to add the spin of the electromagnetic field $S=1$ to the orbital angular momentum $L$ to get the electromagnetic multipoles of definite $J$, which are the vector spherical harmonics. Then any vector-valued function can be resolved into a sum of these harmonics. I don't know of any references that does it this way. Too bad. $\endgroup$ – QuantumDot Jul 1 '16 at 10:37

You can try Modern Problems in Classical Electrodynamics by Charles Brau. I think it is relatively clear in this book.

At least I hope you can find the end-results in Zangwill or Brau. Then you should after trying be able to get there yourself.

Start from the expression for the electric potential. In the integral there is a $\frac{1}{|\bar{r}-\bar{r}'|}$. You just need to make a Taylor expansion of that where $\bar{r}'$ is small. Then you put that term for term into the integral and integrate. You will then get the multipole expansion for the electric potential. Taking the gradient times a minus sign will give you the electric field.


The Taylor series expansion along with Spherical wave expansion is presented in the book "Theory of Electromagnetic Wave Propagation" by Papas.


The three dimensional Taylor expansion of the scalar and vector potentials due to a monochromatic current density is used to derive the Electric and magnetic fields. The fields of electric dipole and quadruple are derived in this manner. The expansion uses

$$ \cfrac{e^{ik\left|\mathbf{r}-\mathbf{r}'\right|}}{\left|\mathbf{r}-\mathbf{r}'\right|}=\sum_{n=0}^{\infty}\cfrac{1}{n!}\left(-\mathbf{r}'\cdot\nabla\right)^{n}\cfrac{e^{ikr}}{r} $$

This method is more suited "if the source can be described as a superposition of an electric dipole, a magnetic dipole and an electric quadrupole"

Also, the spherical harmonic expansion given in Jackson was proposed by "Casimir and Bouwkamp" in their paper, which has elaborate presentation.

  • $\begingroup$ "Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. [...] Answers containing only a reference to a book or paper will be removed!" $\endgroup$ – Chris Feb 7 '18 at 3:49
  • $\begingroup$ @Chris The OP asks "any other reference doing the same expansion, in some other manner", not for a book review. $\endgroup$ – my2cts Sep 9 '20 at 6:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.