Torque on a Current Loop — Question 10

Question 10

Internal source resistance makes torque approach a turn-count saturation limit

A coil has N turns of fixed area A, each adding resistance R1. A source of emf E and internal resistance r drives it. The field B and normal angle theta are fixed, with positive sine. Treat N as continuous for optimization.

1. Derive torque as a function of N and its large-N limit.

2. Find the N that gives half the limiting torque. Show that the coil heating is maximal at the same N.

Question 10 - Solution

Internal source resistance makes torque approach a turn-count saturation limit

(a) Method and calculation

$$I=E/(r+NR_1),\quad\tau(N)=\frac{NEAB\sin\theta}{r+NR_1}.$$$$\tau_\infty=\frac{EAB\sin\theta}{R_1},\quad\tau/\tau_\infty=\frac{NR_1}{r+NR_1}.$$

(b) Interpretation and further calculation

$$\tau=\tfrac12\tau_\infty\Rightarrow NR_1=r\Rightarrow N=r/R_1.$$$$P_{\mathrm{coil}}=\frac{E^2NR_1}{(r+NR_1)^2},\quad\frac{dP_{\mathrm{coil}}}{dN}=\frac{E^2R_1(r-NR_1)}{(r+NR_1)^3};\quad P_{\max}\text{ at }NR_1=r.$$

Physical check

Increasing turns helps until source resistance becomes negligible compared with coil resistance; the ideal-voltage limit recovers turn-independent torque.