Force of a solenoid on a ferromagnetic material

As I was trying to find an answer to another problem, I was informed that the equation I was using would not work.

The equation I was using was
$$F = (NI)^2mu_0frac{text{Area}}{2g^2}$$

If this is indeed wrong, could someone please explain how it is wrong? I have spent some time trying to figure out what’s wrong, and I can’t seem to find the issue.

It seems to me, like my only problem is the one I asked about in my original question.

TIA, to anyone who can clarify this problem.

Edit:
I may have just solved my own problem. I think I may have written the equation in my other question wrong. Looking at it, I don’t know how I missed this before, it looks like I wrote it as $$F = (NI)mu_0frac{text{Area}}{2g^2}$$ and not the correct equation of $$F = (NI)^2mu_0frac{text{Area}}{2g^2}$$
While I believe this is correct, could someone please verify?
I am going to be rearranging it to find the number of coils I require in my solenoid, and I really don’t want to get this wrong.

Edit:
So, in order to help this along, I am going to do an example. As an aside, I was asking my other question because I was getting some funny numbers out of this when I used a value of Area taken from plane b.

Setup:

So, if I am presented with a cylinder 25mm long, and 5mm around, and I am asked to find the number of turns a solenoid that exerts a total force of 1 Newton/second on the cylinder at a distance of 5mm, and that uses 3 Amperes current must have, I would start by rearranging my solenoid force equation.

$$
begin{align}
F &= (NI)^2mu_0frac{text{Area}}{2g^2} [10px]
%
frac{F}{mu_0} &= (NI)^2frac{text{Area}}{2g^2} [10px]
%
frac{F}{mu_0frac{text{Area}}{2g^2}} &= (NI)^2 [10px]
%
sqrt{frac{F}{mu_0frac{text{Area}}{2g^2}}} &= NI [10px]
%
frac{1}{I}sqrt{frac{F}{mu_0frac{text{Area}}{2g^2}}} &= N [10px]
end{align}
$$

Where F is the force exerted on the cylinder in Newtons, N is the number of turns, I is the current passing through the coil, μ₀ is 4π x 10^-7, Area is is the area of the cylinder in plane b(See photo), and g is the distance separating. the cylinder from the coil.

Example of planes passing through a cylinder:
$hspace{175px}$.

$$
begin{align}
sqrt{frac{(1)}{(4π · 10^{-7})frac{text{Area}}{2(15)^2}}}/(3) &= N [10px]
sqrt{frac{1}{4π · 10^{-7}·frac{text{(19.634375)}}{2(15)^2}}}/3 &= N [10px]
end{align}
$$

And we know area is about $19.634375$ because area is $pi r^2$ and the radius was $2.5 , mathrm{mm}$.

$$
begin{align}
sqrt{frac{1}{4π · 10^{-7}· 19.634375}}/3 &= N [10px]
sqrt{frac{1}{0.000024673283}}/3 &= N [10px]
sqrt{40529.66927830399}/3 &= N [10px]
201.3198183942753/3 &= N [10px]
N &= 67.1066061314251 [10px]
N &= 67.1~text{Turns}
end{align}
$$

I don’t believe I missed anything but if I did something wrong, please tell me. I hope this helps determine if this is the correct equation or not.

Edit:
I don’t suppose anyone can check my math?