Influence of coil geometry and coil-plasma distance on the magnetic field approximation error

We investigate analytically two questions: 1) How does the coil geometry influence the effect of electric current noise on the induced magnetic field? 2) How does the coil-plasma distance influence our ability to control the pointwise magnetic field error in terms of the average magnetic field error? Regarding (1), we argue that the main geometric quantities of interest are the notion of reach and the volume of the region enclosed by the coils. Our main finding is a quantitative formula which shows that the larger the reach and the smaller the volume of the region enclosed by the coils, the smaller is the influence of the electric current uncertainty on the magnetic fields. Regarding (2), we show that the pointwise magnetic field error can be controlled (modulo an explicit constant) by the average square-magnetic field-error times $(\text{coil-plasma distance}{)}^{-\frac{3}{2}}$.