Question

A wire is formed into a circle having a diameter of 13.0 cm and placed in a uniform magnetic field of 2.50 mT. The wire carries a current of 5.00 A.
(a) Find the maximum torque on the wire.
_____ µN·m
(b) Find the range of potential energies of the wire-field system for different orientations of the circle.
minimum
_____µJ
maximum
_____ µJ

Answer 1

\(\begin{aligned} \text { diameter, } d &=13.0 \mathrm{~cm} \\ &=0.13 \mathrm{~m} \end{aligned}\)

area of the coil, \(A=\frac{\pi d^{2}}{4}\) \(=\frac{\pi(0.13 \mathrm{~m})^{2}}{4}\)

\(=0.0133 \mathrm{~m}^{2}\)

magnetic field, \(B=2.50 \mathrm{mT}\) \(=2.50 \times 10^{-3} \mathrm{~T}\)

current, \(I=5.00 \mathrm{~A}\)

(a)

maximum torque:

\(\begin{aligned} \tau &=B L A \\ &=\left(2.50 \times 10^{-3} \mathrm{~T}\right)(5.00 \mathrm{~A})\left(0.0133 \mathrm{~m}^{2}\right) \\ &=166 \times 10^{-6} \mathrm{~N} \cdot \mathrm{m} \\ &=166 \mu \mathrm{N} \cdot \mathrm{m} \end{aligned}\)

(b) potential energy:

\(\begin{aligned} U=& L A B \cos \theta \\ U_{\min } &=-\angle A B \\ &=-(5.00 \mathrm{~A})\left(0.0133 \mathrm{~m}^{2}\right)\left(2.50 \times 10^{-3} \mathrm{~T}\right) \\ &=-166 \times 10^{-6} \mathrm{~J} \\ &=-166 \mu \mathrm{J} \\ U_{\max } &=\angle A B \\ &=(5.00 \mathrm{~A})\left(0.0133 \mathrm{~m}^{2}\right)\left(2.50 \times 10^{-3} \mathrm{~T}\right) \\ &=166 \times 10^{-6} \mathrm{~J} \\ &=166 \mu \mathrm{J} \end{aligned}\)