# Repulsive Force between two Electromagnets

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

1. they attract each other and are pulled apart or
2. 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.

• Wikipedia has a long list of examples of such calculations. – Winther Sep 10 '15 at 18:53
• Sebastian Riese gave you a good pointer there, but unfortunately it's not going to work for scenarios with ferromagnetic materials, which you will need, if you want to make this work. You could use a simulation program like MaxFem for the calculations, but there is a rather steep learning curve associated with that. As a practical pointer: start with permanent magnets and figure out a way to use a weak electromagnet just for stability control. Do not try to create the load bearing magnetic field with an electromagnet, cheap electromagnets don't have the power density of permanent magnets. – CuriousOne Sep 10 '15 at 20:20
• @Winther: My first site to visit was Wikipedia but it did not specify for electromagnets. – Gouthamm4G Sep 11 '15 at 13:48
• @CuriousOne: Since I already have the materials for making an electromagnet, I didn't want to use permenant magnets. Besides, if I use electromagnet I could easily vary the magnetic field in real time for future changes. – Gouthamm4G Sep 11 '15 at 13:51
• And you will need those material to control the levitation. All I am saying is that you will find it much easier to do if you start with permanent magnets to create >90% of the required forces. You don't have to do it this way, it just turns out to be the by far better approach. – CuriousOne Sep 11 '15 at 13:57

$$\vec F = I \oint_{\gamma_2} d\vec r \times \vec B(\vec r).$$
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: $$\vec B(\vec r) = \frac{\mu_0 I}{4\pi} \int_{\gamma_1} \frac{d\vec r' \times (\vec r - \vec r')}{\left| \vec r - \vec r' \right|^3}.$$