When a wire or a coil of wires is carrying a current there will be a circular magnetic field around it. If that wire or coil is then put into a magnetic field between a North and South pole the two magnetic fields interact like this:

The two fields together cause a force on the wire and logically (using Newtons Third Law) an equal and opposite force on the magnets. Looking at a side view of the magnet the magnetic field below the wire is reinforced whereas the two fields above the wire oppose each other so the overall field is weak. The wire is then pushed up towards the weaker field and the magnet is pushed down.

The size of this force can be measured using a setup like the diagram below:

The downward force is measured in grams on a mass balance, which is then converted to Newtons. The size of the force depends on the strength of the magnetic fields, the number of wires in the coil, the length of the coil in between the poles of the magnet and on the current flowing through the coil. In this setup the force F=NBIL where F is the force in newtons, N is the number of turns in the coil, I is the current, L the length of the coil between the magnetic poles and B the magnetic flux density of the fixed magnets.

We rearrange that to find the flux density of the field of the large magnet, so B = F/NIL

The video below shows the experiment performed and measurements taken

There is a PDF of notes to download which extends the examples of forces in a magnetic field

Forces in a magnetic field