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

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?

• In linear media (i.e., when you can assume vacuum permittivity and permeability), then the superposition of all the sources works great. Part of the problem is that you measure the total field so without other information, you cannot determine much about the individual sources from a single-point measurement. May 19, 2017 at 20:33

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.

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

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.