Calculating the Current in a Single-Loop Circuit — Question 7

Question 7

Both supply wires contribute to the loop resistance

An ideal source E drives a load R through two identical round leads, each length L, radius a and resistivity rho. Contact resistance is negligible.

1. Derive the current including the leads.

2. Find the minimum radius needed to keep current at least fraction f of the ideal-lead value E/R, with $0<f<1$.

Question 7 - Solution

Both supply wires contribute to the loop resistance

(a) Method and calculation

$$R_w=2\rho L/(\pi a^2).$$$$I=\frac E{R+2\rho L/(\pi a^2)},\quad I_{\mathrm{ideal}}=E/R.$$

(b) Interpretation and further calculation

$$\frac I{I_{\mathrm{ideal}}}=\frac R{R+R_w}\ge f\Rightarrow R_w\le R(1-f)/f.$$$$a\ge\sqrt{\frac{2\rho Lf}{\pi R(1-f)}}.$$

Physical check

The outgoing and return leads carry the same current and their resistances add.