Why is the SI unit of Magnetic Pole Strength Ampere-meter?

Question

Till now I was silent because I thought the SI unit of pole strength to be Ampere-Second and due to everything in analogy with electrostatic, it was ok, but when I looked closely, it changed to Ampere Meter, so my question is the expression at that time,WHY?

Emilio Pisanty · Accepted Answer

First off: there are no magnetic (mono)poles. As such, specifying units for quantities that don't exist is a rather futile (and not particularly well-defined) exercise.
That said, if there were magnetic monopoles, then you would expect that if you had two opposite magnetic monopoles of strength $pm m$ separated by a distance $d$, the result would be a magnetic dipole of strength $mu = md$.
And, luckily, magnetic dipoles do exist (they are basically loops of electric current), and their units are known (specifically, the units are $rm A:m^2$ and the dimensions are $[mu]=[IL^2]$, corresponding to the strength of the electric current times the area of the loop).
This fixes the dimensions of $m$, since we would require $[md]=[IL^2]$ and therefore $[m]=[IL]$.

Davide Dal Bosco · Answer

I never came across this quantity, but I read that one can derive the units of pole strength by assuming that the magnetic force to have a form similar to Coulomb's Law:
begin{equation}
F = frac{mu_0}{4pi}frac{m_1 m_2}{r^2}
end{equation}
where $m_1$ and $m_2$ are the magnetic (mono)poles and act as the charges in Coulomb's Law.
As we know that:

The force $F$ is measured in newton ($mathrm{N}$);
The permeability $mu_0$ is measured in newton per ampere squared ($mathrm{N} / mathrm{A}^2$);
The distance $r$ is measured in meters ($mathrm{m}$);

we must have that $m_1$ and $m_2$ have the units of ampere times meter ($mathrm{A}cdotmathrm{m}$) to guarantee the dimensional soundness of the equation.
Indeed, from the previous equation
begin{equation}
[mathrm{N}] = frac{[mathrm{N}]}{[mathrm{A}]^2}frac{[m_1] [m_2]}{[mathrm{m}]^2}quadrightarrowquad [m_1] = [m_1] = [mathrm{A}]cdot[mathrm{m}]
end{equation}
PS. recall that the Coulomb's law reads $F = frac{1}{4piepsilon_0}frac{q_1 q_2}{r^2}$.

Why is the SI unit of Magnetic Pole Strength Ampere-meter?

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