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As I understand it, the field strength at the midpoint of two Helmholtz coils is given by:

$$B=\left(\frac{4}{5}\right)^{3/2} \frac{\mu_0 nI}{R}$$

All things being equal, it seems that we can control the field strength by controlling the current through the coils. This would imply that all one needs in order to create an arbitrary field strength is a current-control system. Intuition would lead me to believe that you would need some sort of magnetometer feedback control system instead.

Is this equation an idealized approximation with no external magnetic fields, or is this true even in the presence of external fields?

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The equation is rigorously valid only if the coil with its current is the only object producing a magnetic field. If you have other origins of magnetic fields, the formula gives you the increase of $B$ due to the current flowing in the coil.

I confirm that, if you want to impose a given value of magnetic field in a point, where there are already other magnetic fields, then you need something to measure them; then you can apply a $B$ which is the difference between what you have and what you would like to get.

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    $\begingroup$ More generally all instances of "the field at [location] relative [thing]" refer to the field contribution due to the [thing], and the total field always consists of the vector sum of all contributions. $\endgroup$ Jul 26, 2019 at 15:52

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