Torque on a Current Loop — Question 5

Question 5

Adding turns at fixed voltage can leave torque unchanged when wire resistance also grows

A coil has N identical turns, each of area A and perimeter p. The wire has resistivity rho and cross-sectional area s. An ideal voltage source V drives it. The magnetic field and the coil-normal angle theta are fixed; neglect lead resistance.

1. Find the coil current and magnetic torque as functions of N.

2. Compare doubling N under fixed voltage with doubling N under a maintained fixed current.

Question 5 - Solution

Adding turns at fixed voltage can leave torque unchanged when wire resistance also grows

(a) Method and calculation

$$R_N=\rho Np/s,\quad I_N=V/R_N=Vs/(\rho Np).$$$$\tau_N=NI_NAB\sin\theta=\frac{VsAB\sin\theta}{\rho p}.$$

(b) Interpretation and further calculation

$$V\text{ fixed}:\quad I_{2N}=I_N/2,\quad\tau_{2N}=\tau_N.$$$$I\text{ fixed}:\quad\tau_{2N}=2\tau_N.$$

Physical check

Turn count alone does not determine torque; the electrical driving constraint matters.