# Derivation for Magnetic Field Due to Dipole

## Main Question or Discussion Point

I am looking for a reference (derivation) for this exact formula given for "the magnetic field due to a dipole $$\mu$$ fixed at the origin" :

$$B=-\frac{\mu}{R^3}+\frac{3r(\mu\cdot r)}{R^5}$$

I don't really know anything about dipoles or how they are derived (I have only taken lower division E&M) so the $$\mu$$ has no meaning to me. It seems that this formula is derived from a vector potential $$A=\frac{\mu\times r}{r^3}$$ and I know that B = grad x A so it might help more to understand what this potential actually means as well as a good lesson on what is the dipole vector mu. Thanks.

note:i'm having trouble rendering the tex so here is the formula in plain text:

B=-u\R^3 + (3r(u dot r))\R^5 here i use small r for the position vector and R for the length of r and u stands for the greek letter mu for the dipole vector.

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## Answers and Replies

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jtbell
Mentor
I know it's in Griffiths's "Introduction to Electrodynamics." It's probably in other intermediate-level E&M textbooks such as Purcell or Lorrain & Corson.

Griffiths does it as part of the general "multipole expansion" of the magnetic vector potential from a general distribution of current. You might try a Google search for something like "magnetic vector potential multipole expansion."