How do you calculate the resulting magnetic field for multiple sources?

Question

I've been looking at some fusion reactors and I keep wondering how putting some kind of extra magnet in some configuration would affect the field, but I don't know how to figure it out. Like for example if you took a solenoid and sat a permanent magnet down next to it. This is the only thing I could find: https://www.quora.com/How-do-I-calculate-the-magnetic-field-created-by-a-number-of-magnets

Edit: I'm pretty sure it's just super positioning, but I need someone else to answer, because I don't know for sure. Also does it follow from Maxwells equations?

Answer 1

Since the Maxwell's equations are linear partial differential equations, you can compute the magnetic field due to multiple sources by superposition.

A really important application relies on the superposition principle for magnetic fields is the Biot–Savart law i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.

                                              

$$\mathrm d\vec B = \frac{\mu_0}{4\pi}\frac{I \; \mathrm d\vec l \times \vec R}{R^3}$$

Answer 2

You are correct, they follow superposition. The magnetic field is a vector field, and so they follow a vector sum when they are in a superposition.

Maxwell's equations are linear ($\nabla \times$ and $\nabla \dot{}$ are linear operators) and it follows that solutions ($E$ and $B$) obey the superposition principle.