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This is a continuation of another post where I was trying to understand the basic principles regarding the calculation of the attractive force between a magnetized material and an object with infinite permeability (i.e. ferromagnetic material). At the end I understood that the situation can either be seen as a problem of induced magnetization or as a case of mirror magnetic ficticious charges between the magnetized material and the ferromagnetic surface.

To apply this and so others can benefit from this insight, I'm working on a problem of a straight cylinder with uniform magnetization along its axis, which is placed over one of its flat faces on a ferromagnetic surface and thus experiments an attractive force towards it. To approach this problem, I did the following approximation.

In Zangwill's book for electrodynamics, there's a similar exercise in page 435 that describes the coulum-like force betweem two coaxial cylindrical rods with constant magnetizations, which are essentially replaced by for equivalent ficticious magnetic charges at their extremes in the configuration (-)(+) (+)(-).

Thus my idea was to replace the cylinder with two equivalent cylindrical rings on its bases, placing two mirror charges beneath the ferromagnetic surface, and calculating the force between the rings with an associated surface current which can be easily calculated from the given magnetization (it's worth noting that there are no volumetric charges as $\nabla \times \vec{M}=0$ for a constant magnetization).

My exact question is, is it valid to use this image method approach for cases like this, or to even consider the forces of mirror rings of current, or is there an assumption I am missing?

Attempt at solution

In case you want further details, I include some of my latest calculations.

The vector potential of a filametary current ring of radius R is an known result given in Zangwill (p. 325):

$$\vec{A(\rho,z)}=\vec{\phi}\frac{\mu_0 IR}{2}\int_{0}^{\infty} (dk J_1(k\rho)J_1(kR))e^{-k|z|}$$

where $J$ and $J_1$ are elliptic integrals. Now the force between two rings separated a distance $h$ can be calculated using mutual inductance $L_M$, defined as:

$$L_M=\frac{1}{I^2}\int d^3 r \vec{j_1(r)}\cdot\vec{A_2(r)}$$

Substituting

$$L_M=\mu_0 \pi R^2 \int_{0}^{\infty} dk J_1(kR)exp(-kh)$$

And the force is:

$$\vec{F}=I^2 \frac{dL_M}{dh}\vec{z}=\mu_0 \pi I^2 R^2 \frac{d}{dh}\int_{0}^{\infty} dk J_1(kR)exp(-kh)$$

Now if we consider the case of mirror magnetic charges in the configuration (-)(+) (+)(-), we get:

$$F_{tot}=F_{(-)(-),1}(h=2L)+F_{(-)(+),2}(h=L)+F_{(+)(+),3}(h=0)$$

However, the correct answer I found is:

$$F=8\pi a^2 L M^2\left [\frac{K(k)-E(k)}{k}-\frac{K(k_1)-E(k_1)}{k_1}\right ]$$

So there's a clear contradiction in my procedure for in the calculation of forces (there should only be two terms in the first place). The $K$ and $E$ are the elliptical integrals.

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