In the problem below, why does the second method used to calculate magnetic dipole moment $\vec {\mu}$ essentially use the definition of the electric dipole moment $\vec {p}$? Is magnetic dipole moment the same as electric dipole moment? Can this method always be used? If yes, what is the need to differentiate between the two?

enter image description here


Electric dipole moment and magnetic dipole moment are not the same quantity, because (of course) an electric dipole moment creates an electric field, while a magnetic dipole moment creates a magnetic field.

However, in the presence of magnetized matter and in the absence of free currents, it is possible to define a (fictitious) "magnetic charge" $\rho_m = -\vec{\nabla} \cdot \vec{M}$. (On the boundary between media, we can similarly define $\sigma_m = \vec{M} \cdot \hat{n}$.) The auxiliary field $\vec{H}$ then satisfies $\vec{\nabla} \cdot \vec{H} = \rho_m$ and $\vec{\nabla} \times \vec{H} = 0$, so we can use the same mathematical techniques to find $\vec{H}$ as we do to find the electric field outside a known charge distribution.

In particular, this allows us to define a "Coulomb's Law" for $\vec{H}$, in analogy to that for the electric field: $$ \vec{H}(\vec{r}) = \frac{1}{4 \pi} \iiint \rho_m(\vec{r}')\frac{\vec{r} - \vec{r}'}{|\vec{r} - \vec{r}'|^3} \, d^3\vec{r}'. $$ One can then perform a multipole expansion for this $\vec{H}$ in powers of $r^{-1}$; and the "dipole term" for this expansion is defined in terms of $\rho_m$ in exactly the same way that $\vec{p}$ is related to $\rho$.

For more information on this technique, I recommend Zangwill's Modern Electrodynamics, as well as my answers here and here. Or, perhaps, "Slide 5" from your instructor's notes.

  • $\begingroup$ Thank you! Slide 5 didn't help but your links did. $\endgroup$
    – dimes
    Oct 16 '20 at 19:49
  • $\begingroup$ @dimes Can you please share the link to your instructor's notes slides? $\endgroup$ Oct 2 '21 at 8:02

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.