Torque on a Current Loop — Question 4

Question 4

The best rectangular loop uses its available wire length to maximize area

A single-turn rectangular loop uses a fixed wire length P and carries a maintained current I in a uniform field B. Its orientation is chosen for maximum torque. Neglect the lead length.

1. Find the rectangle dimensions that maximize torque and give the maximum value.

2. Compare this value with a circular loop made from the same wire at the same current and orientation.

Question 4 - Solution

The best rectangular loop uses its available wire length to maximize area

(a) Method and calculation

$$2(a+b)=P\Rightarrow A=a(P/2-a),\quad dA/da=P/2-2a.$$$$a=b=P/4,\quad A_{\max}=P^2/16,\quad\tau_{\mathrm{square}}=IBP^2/16.$$

(b) Interpretation and further calculation

$$2\pi r=P\Rightarrow A_{\mathrm{circle}}=P^2/(4\pi),\quad\tau_{\mathrm{circle}}=IBP^2/(4\pi).$$$$\tau_{\mathrm{circle}}/\tau_{\mathrm{square}}=4/\pi\simeq1.27324.$$

Physical check

For a given perimeter, the circle encloses more area than any rectangle; the comparison assumes equal maintained current.