Imagine a solenoid which has current $i$ and is producing a magnetic field $B$ which equals $$B=\mu N i$$
Now, imagine we put a small cylindrical magnet at the end of the solenoid.
Then because at end of the solenoid, field lines start diverging and are not parallel, a force is exerted on the magnet from the magnetic field created by the solenoid.
What is the magnitude of this force?
Best Answer
Depends on the orientation of the magnet at the end of the solenoid as to how strong the force is.
This is the equation that governs how the current in the solenoid creates the magnetic field around it
$ \nabla \times B = \mu_0 j $ where j is the current density.
This equation shows the potential that can be created from a magnetic field on something charged
$ U = \mu B $ where $\mu$ is the dipole moment.
for using $ F = \frac{dU}{dx} $
so we can say that $ F = \mu \frac{\partial B}{\partial x} $
Similarly, the magnetization is the net magnetic dipole moment in the material:
$M = N \mu $ or $ M $ ~ $ \mu $
therefore the force exerted would be:
$ F = M \frac{\partial B}{\partial x} $
This can be shown in more detail using the material versions of Maxwell's equations, but the main idea behind it is that a magnetic field B acts on the magnetic dipole moments of another magnetic material.