A Current-Carrying Coil as a Magnetic Dipole — Question 10

Question 10

The best coil radius depends on whether turns or wire length is held fixed

A thin circular coil must maximize the field at a prescribed axial distance z>0. Its radius R is adjustable and $B=\mu_0NIR^2/[2(R^2+z^2)^{3/2}]$. Ignore leads, heating and winding thickness. Current I remains fixed. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the optimal R when N is fixed.

2. Instead hold total wire length Lw fixed, with $N=L_w/(2\pi R)$ treated as continuous. Find the new optimal R and compare the design constraints.

Question 10 - Solution

The best coil radius depends on whether turns or wire length is held fixed

(a) Method and calculation

$$\frac d{dR}\ln B=\frac2R-\frac{3R}{R^2+z^2}=0\Rightarrow2z^2-R^2=0.$$$$\boxed{R_{\mathrm{opt}}=\sqrt2z}\quad(N,I\text{ fixed});\quad B\to0\text{ as }R\to0\text{ or }\infty.$$

(b) Interpretation and further calculation

$$B=\frac{\mu_0IL_wR}{4\pi(R^2+z^2)^{3/2}},\quad\frac d{dR}\ln B=\frac1R-\frac{3R}{R^2+z^2}=0.$$$$\boxed{R_{\mathrm{opt}}=z/\sqrt2}\quad(L_w,I\text{ fixed}).$$The wire-length constraint reduces the available turns as the radius grows; an actual winding must also satisfy integer-turn and geometric constraints.

Physical check

Both stationary points are maxima under their respective constraints. Optimizing a field without stating what is held fixed gives an incomplete design problem.