Imagine that I use a long wire to create an electromagnet. Let's also assume that the current flowing along the wire is constant, and that the wire is winded on the vacuumm.
Is the magnetic field generated by this electromagnet (current flowing over an arbitrary path on a finite volume) proportional to the current?
The answer is positive. This is due to the fact that the equations describing how currents generate the field are linear. The solution is obtained by a suitable inverse of the linear operator associating currents to fields. It is fundamental to observe that this inverse operator is linear because the boundary conditions satisfy the superposition principle (this is not obvious but it is true in this case where the boundary conditions are those in the vacuum). Therefore there is a relation like this $$\vec{B}(\vec{x}) = L_{\vec {x}} I$$ where $L_{\vec {x}}$ is a linear operator depending on the point $\vec{x}$ where the filed is evaluated and $I$ the constant current generating the field. You see that if $I$ is replaced by $cI$ the field becomes $c\vec{B}(\vec{x})$ for every constant $c\in \mathbb R$.
