# How do magnetic fields combine?

How do the two fields interact to give the combined field, do they superpose like in waves? And how does this field cause the force on the conductor?

Magnetic fields indeed simply combine via superposition - like in waves. The potential energy of magnetism is $$V=-\frac{1}{2}\mu\cdot B$$ where $$\mu$$ is the magnetic moment (vector) of the, in this case, conductor and $$B$$ is the magnetic field strength (vector) and the dot represents the scalar product. The magnetic moment depends in general on the imposing magnetic field (for para- or diamagnetism) or in the current in the conductor. The force is then simply the negative gradient of the potential, i.e., $$F=-\nabla V=\frac{1}{2}\nabla(\mu\cdot B)$$. The gradient points into the direction of the steepest slope of the function it acts on, in this case the potential $$V$$. As $$V$$ consists of a dot product, we have on the side where the vector arrows point into the same direction a positive value and on the other side a negative value. (On the top and bottom the arrows are perpendicular to each other and the product therefore vanishes at these points and we have a smooth change between zero and the largest value.) The gradient now points into the direction where we have the larger value (where the vector arrows are parallel and not anti-parallel) but since we have for the force the negative gradient the resulting force points outwards of the magnet in your sketch.