In a current-free region, we have $\nabla \times \mathbf{B} = 0$, allowing us to write $\mathbf{B} = -\nabla V$ for some scalar function $V$. We also have $\nabla \cdot \mathbf{B} = 0$, meaning that the function $V$ must satisfy the Laplace equation:
$$
\nabla^2 V = 0
$$
The solution to this equation is well known in spherical coordinates, and it is given by (in complex form),
$$
V(r,\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l A_l^m \frac{1}{r^{l+1}} Y_l^m(\theta,\phi) + B_l^m r^l Y_l^m(\theta,\phi)
$$
where $Y_l^m(\theta,\phi)$ are the spherical harmonics. If we consider only the part of the potential due to internal sources, then
$$
V^{int}(r,\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l A_l^m \frac{1}{r^{l+1}} Y_l^m(\theta,\phi)
$$
Now, many authors throw away the $l=0, m=0$ term, starting the summation at $l=1$, in other words setting $A_0^0 = 0$. One example is Campbell's Introduction to Geomagnetic Fields (2003).
My question is why is it allowed to discard the $l=0$ term? It seems to be related to the absence of magnetic monopoles. However the equation $\nabla \cdot \mathbf{B} = 0$ already accounts for no magnetic monopoles, and so the $l=0$ part of the potential already satisfies this condition. So there must be another reason why authors set $A_0^0 = 0$, however I cannot find a satisfactory explanation.
Best Answer
The equations $\nabla \cdot {\bf B}=0$ and $\nabla \times {\bf B}=0$ imply that ${\bf B}$ can be written as the gradient of a scalar potential $V$ satisfying Laplace's equation in the current-free region. However, the coefficients of the spherical harmonic expansion in a region external to the sources of the magnetic field must match the coefficients of the internal solution at the boundary. Therefore, we cannot have complete freedom in the choice of the boundary conditions for the Laplace equation. Any choice incompatible with the condition $$ \int_{\Sigma} {\bf B} \cdot {\mathrm d} {\bf S}=0, $$ where $\Sigma$ is any surface limiting the region of the sources, must be discarded. Notice that this is a request independent of the conditions in the source-free region, therefore it is not automatically satisfied by the spherical harmonic series. It is easy to check by direct substitution of the series expansion, that such a condition on the boundary implies $A^0_0=0$.