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I'm trying to understand the difference between monopole moment and charge or mass itself. What I found is related to magnetic monopole that is irrelevant.

I want to know in multiple expansion of electric charge or mass, the first term is monopole, the second is diple and etc. Monopole describing the background field. I want to know what is it exactly? My initial charge or mass of the object? Assume I'm computing multiple moments for some object, say, the mass of earth. the monopole should be a sphere and as much as we go further in multipole extension we get a little deformed shape of the sphere and sum of them gives of the shape of the earth?

what is the relation of monopole and earth itself?

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  • $\begingroup$ If you only consider monopole then the field is not related to the shape, because different sized earths (different shape) of same mass result in the same field. Only when you take the field inside the object into account, you will have a relation to the shape. But then it‘s not a monopole anymore... hmmm.... nice question! $\endgroup$ – Hartmut Braun Jul 31 '19 at 10:36
  • $\begingroup$ The multipole expansion applies to a $1/r$ potential with a finite density localized to a particular region, and it's only valid outside that region. So you can't do the expansion at a reference point inside the object. $\endgroup$ – Richter65 Feb 3 at 14:55
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The monopole moment is the total quantity of whatever it is you're considering. For example, if you're looking at a 3-dimensional multipole expansion of the mass of the earth, the monopole moment would be the total mass. That doesn't have anything to do with treating the earth like a sphere. Instead, it's the statement that the total mass of the earth (regardless of its shape) is the same no matter how far or which direction you're looking from.

Mathematically, the monopole moment is the $n=0$ moment, so $$\int{r^n \rho(r) \;d^3(r)} = \int{r^0 \rho(r) \;d^3(r)} = \int{\rho(r) \;d^3(r)} = M$$

The answer to this question gives a good way of thinking about it intuitively.

For mass, there will always be a non-zero monopole moment. But that's not true for other quantities. For example, think about the classic example of a positive charge $q$ and negative charge $-q$ separated by a distance $d$. The total net charge is zero, so the monopole moment is zero, but the dipole moment is not: it is $qd$.

You may have noticed that in this case the dipole moment is independent of any reference point. That's because it's the lowest non-zero moment. In general, moments are a function of a reference point, but the lowest non-zero moment (and only the lowest non-zero moment) is independent of where you set the reference point. So, in the earth example, the monopole moment would be independent of the reference point, but dipole moment (and quadrupole moment, etc.) would not.

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