= Far-field approximation for a current loop
{title2=$\mathbf B\simeq\mu_0[3(\mathbf m\cdot\widehat{\mathbf r})\widehat{\mathbf r}-\mathbf m]/(4\pi r^3)$}
Far outside a localized steady source of size $a$, at distance $r\gg a$, its <multipole expansion> starts with the <magnetic dipole moment> if that moment is nonzero. In <vacuum>, using <SI> units,
$$
\mathbf A_{\mathrm{dip}}(\mathbf r)=\frac{\mu_0}{4\pi}\frac{\mathbf m\times\mathbf r}{r^3},
\qquad
\mathbf B_{\mathrm{dip}}(\mathbf r)=\frac{\mu_0}{4\pi r^3}\left[3(\mathbf m\cdot\widehat{\mathbf r})\widehat{\mathbf r}-\mathbf m\right],
$$
where $\mu_0$ is the <vacuum permeability>, $\widehat{\mathbf r}=\mathbf r/r$, and the <magnetic field> is the <curl> of the <magnetic vector potential>. The <magnetic dipole field> falls as $r^{-3}$, and the <magnetic vector potential> as $r^{-2}$. The <localized steady-current moment identities> remove the putative $r^{-1}$ term in the <magnetic vector potential>. No <magnetic monopole> is needed. If different <current loops> cancel the total <magnetic dipole moment>, a higher term of the <multipole expansion> supplies the leading distant <magnetic field>. <Classical Electromagnetism by Richard Fitzpatrick/Localized Current Distribution>{ref}.
The approximation is not valid at the center of a finite <current loop>. For a circular <current loop> of <radius> $R$ and <electric current> $I$, the exact <on-axis magnetic field of a circular current loop> and the distant approximation are
$$
B_z(z)=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}},
\qquad
B_{z,\mathrm{dip}}(z)=\frac{\mu_0IR^2}{2|z|^3}.
$$
Their ratio is $B_z/B_{z,\mathrm{dip}}=(1+R^2/z^2)^{-3/2}$: at $|z|=10R$, the approximate <magnetic field> exceeds the exact <magnetic field> by about $1.50\%$ relative to the exact value. Near the source, retain its geometry rather than replacing it with a point source specified only by its <magnetic dipole moment>.
Back to article page
