This question might seem a bit too basic(or advanced!), but it is the one to which I'm really desperate to find a simple enough explanation and solution.
Ok, So here's the problem,
I have two identical electromagnets with a field strength $B$ each. Each of radius $r$ and length $l$ with $N$ turns say, how do I calculate the net repulsive/attractive force between them if
- they attract each other and are pulled apart or
- they repel each other and are pushed against each other
I want to levitate one magnet carrying a weight (so its stabilized) above the other electromagnet.
Even if the solution is for different fields it is no problem.
Best Answer
The force is given by:
$$ \vec F = I \oint_{\gamma_2} d\vec r \times \vec B(\vec r).$$
Where $\vec B(\vec r)$ is the field generated by coil 1 and $I$ is the current in coil 2 and the integral is along the curve $\gamma$ which is described by the wires of the coil 2. Note, that there might also be a torque.
Knowing the maximal or homogeneous field $B$ in the interior of the coils does not help much.
The field of the coil is given by the Biot-Savart law: $$ \vec B(\vec r) = \frac{\mu_0 I}{4\pi} \int_{\gamma_1} \frac{d\vec r' \times (\vec r - \vec r')}{\left| \vec r - \vec r' \right|^3}.$$
Also note, that levitating a stationary magnet in a stationary magnetic field is always unstable due to the Earnshaw theorem. You will have to use some form of regulation to keep the system stable. Calculating some fixed parameters will not help you (but might help in the design of the system).