# Force from solenoid

I'd like to approximate the force from a solenoid, or at the very least find a formula which is proportional to the force so that I can experimentally find the constant for my particular case. Apparently an exact answer to this is hard and involves quantum physics that is a bit beyond me; something about the Ising model. I've found $F =\frac{(NI)^2 μ_0A}{2g^2}$, where $N$ is number of loops, $I$ is current though the solenoid, $A$ is cross-sectional area of the inside of the solenoid and $g$ is the "is the length of the gap between the solenoid and a piece of metal". I found that here.

Is this a decent approximation? If so could you more rigorously define $g$, because I don't know what distance this is supposed to be? If it is not, what might I use instead? I found the Gilbert model on Wikipedia that is apparently a good approximation of the force between magnets. Could I use this to approximate the force on a piece of iron from a solenoid?

• I think the model you are referring to would be a good approximation only if the distance between the two objects is much greater than their dimensions. In case of a piece of iron, you would need to know the magnetic dipole moment per unit volume and then use integration and the expression for the force between two dipoles to calculate the resultant force. The calculation would be quite complex though. – guru May 7 '14 at 17:42
• The model you cite assumes very specific and more or less hypothetical assumptions regarding the geometry of the solenoid and its flux path. In real world solenoids this geometry is purposely complex to maximize magnetic force or to provide uniform force over the travel of the solenoid. The Ising model, and similar models can only provide a rough estimate of the force-current-displacement behavior. For more practical and accurate predictions, requires numerical methods are needed - 3D finite electromagnetic modeling software. – docscience Feb 16 '15 at 3:55

The interior of a solenoid the magnetic field is uniform in magnitude and direction and is given by: $$B=\mu_0 \frac{NI}{L}$$ Where $N$ is the number of turns, $L$ the length of the solenoid and $I$ the current through it. Applying $$F=qvB\sin\theta$$ the force can be found.