A Helmholtz coil system consists of two identical coaxial circular loops of radius \(R\), each carrying a steady current \(I\) in the same direction, separated by an adjustable distance \(s\). The axial magnetic field produced by a single loop centered at the origin is given by \(B_1(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\). By symmetry, the first derivative \(\left.\dfrac{dB}{dz}\right|_{z=0}\) is zero at the midpoint \(z = 0\) between the coils for any separation \(s\). To achieve maximum field uniformity near the midpoint, the second derivative must also vanish, such that \(\left.\dfrac{d^2B}{dz^2}\right|_{z=0} = 0\).

What coil separation \(s\) satisfies this uniformity condition, and what is the resulting net magnetic field magnitude \(B(0)\) at the midpoint?