Solenoids and Toroids — Question 5

Question 5

A fixed copper volume and heating budget remove the apparent advantage of more turns

A long thin solenoid has fixed radius R, length ell, copper resistivity rho and copper volume Vc. It is wound with N turns; ignore lead length and insulation volume. The wire cross section is adjusted as N changes. The allowed steady resistive power is P. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Derive the wire resistance and the current allowed by P.

2. Show how the maximum ideal center field depends on N and explain the tradeoff in wire gauge.

Question 5 - Solution

A fixed copper volume and heating budget remove the apparent advantage of more turns

(a) Method and calculation

$$L_w=2\pi RN,\quad A_w=\frac{V_c}{L_w},\quad\mathcal R=\rho\frac{L_w}{A_w}=\frac{\rho(2\pi RN)^2}{V_c}.$$$$\boxed{I=\sqrt{P/\mathcal R}=\frac1{2\pi RN}\sqrt{PV_c/\rho}}.$$

(b) Interpretation and further calculation

$$B=\frac{\mu_0NI}{\ell}=\boxed{\frac{\mu_0}{2\pi R\ell}\sqrt{\frac{PV_c}{\rho}}}.$$$$N\to2N:\quad A_w\to A_w/2,\ \mathcal R\to4\mathcal R,\ I\to I/2,\ B\to B.$$

Physical check

The N independence assumes unchanged material resistivity, copper volume and permitted power; it does not include thermal or packing changes.