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
$$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$
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 $$F = (NI)\mu_0\frac{\text{Area}}{2g^2}$$ and not the correct equation of $$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$
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.
$$
\begin{align}
F &= (NI)^2\mu_0\frac{\text{Area}}{2g^2} \\[10px]
%
\frac{F}{\mu_0} &= (NI)^2\frac{\text{Area}}{2g^2} \\[10px]
%
\frac{F}{\mu_0\frac{\text{Area}}{2g^2}} &= (NI)^2 \\[10px]
%
\sqrt{\frac{F}{\mu_0\frac{\text{Area}}{2g^2}}} &= NI \\[10px]
%
\frac{1}{I}\sqrt{\frac{F}{\mu_0\frac{\text{Area}}{2g^2}}} &= N \\[10px]
\end{align}
$$
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?
Best Answer
The force on a magnetic moment will depend on the field gradient. For a magnetic dipole on axis with a thin coil, the problem is fairly easy to solve. (Assume axis is in the z-direction.)
Calculate the $B$ field from a thin coil as a function of $z$.
Differentiate with respect to axis direction ($z$) to find the field gradient.
Calculate force, $F = u\frac{dBz}{dz}$.
The force will depend on the distance between the coil and magnetic moment.
(A magnetic moment in a uniform $B$ field experiences no force - only a torque.)