Force between poles of two magnets

[Meow12]

Homework Statement

What is the magnitude of the force between two magnets if their poles are separated by a distance ##r##? What is the formula and how would you derive it?

Relevant Equations

Biot-Savart law: ##\displaystyle d\vec B=\frac{I d\vec l\times\hat r}{r^2}##

Lorentz force law: ##\vec F = q\vec v\times\vec B##

The Biot-Savart law gives us the magnetic field created by an electric current. We can calculate the force exerted on a moving charge by this magnetic field using Lorentz's force formula.

But my textbook doesn't address the repulsive or attractive force between like or unlike poles (respectively) of two magnets.

 

[Orodruin]

The potential energy of a magnetic dipole ##\vec m## in a magnetic field ##\vec B## is given by
$$
U = - \vec m \cdot \vec B
$$
Use this and take derivatives as usual to find forces and torques .