Exercise 4.1.20: Flat spiral (pancake) coil

The ring between ${\rho }^{\prime }$ and ${\rho }^{\prime }+d{\rho }^{\prime }$ carries $dI=Kd{\rho }^{\prime }=\frac{NI}{b}d{\rho }^{\prime }$ and gives $d{H}_z=dI/2{\rho }^{\prime }$ at its centre:

With the on-axis loop formula,

which reduces to $\ln 2$ for $h\to 0$. The bracket equals $0.375$ at $h=b$ and $0.140$ at $h=2b$:

Twenty turns at $\rho =b$ give $H=NI/2b=6.67$ kA/m: spreading the turns out to $2b$ lowers the centre field by the factor $\ln 2=0.69$, but the field falls off more slowly with $h$ (a single loop gives $0.35H(0)$ at $h=b$, the spiral $0.54H(0)$).

Units: $NI/b$ is A/m. Because $d{H}_z=dI/2{\rho }^{\prime }$, the inner turns are worth most at the centre, and the outer turns add less field but extend it further along the axis (Figure 4.1.33): a wireless-charging pad wants a flat field over the receiver, so its turns are spread outward (Application 4.1.4). At $100$ to $200$ kHz the same coil is run with a few amperes and the field is a few hundred microteslas, enough to transfer several watts across a few millimetres.