Torque on a Current Loop in a Uniform Field: 17-Phys-A3 Electromagnetics, December 2016, Q7

Question 7: Torque on a Current Loop in a Uniform Field (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Square loop, side $10$ cm, $N=10$ turns, $I=1$ A, lying in an east–west VERTICAL plane; viewed from due north, the current circulates clockwise; ambient field $B=0.1$ T, pointing straight up.

By the right-hand rule, current clockwise as viewed from the north gives a magnetic moment pointing south; $\boldsymbol\tau=\mathbf m\times\mathbf B$ then points due west.

Find. Magnitude and direction of the torque on the loop.

Approach. Find the loop's magnetic moment direction from the right-hand rule applied to the stated (clockwise-from-north) current sense, then compute $\boldsymbol\tau=\mathbf m\times\mathbf B$.

Magnetic moment direction. Current clockwise as seen by an observer standing to the north (looking south) curls, by the right-hand rule, to a moment pointing AWAY from that observer — i.e. due south. Its magnitude: $$m=NIA=(10)(1)(0.10)^2=\boxed{0.10\ \text{A}\cdot\text{m}^2}.$$
Torque magnitude. $\mathbf m$ (south, horizontal) is perpendicular to $\mathbf B$ (up, vertical), so $\sin\theta=1$: $$\tau=mB\sin\theta=(0.10)(0.1)(1)=\boxed{0.010\ \text{N}\cdot\text{m}}.$$
Torque direction. With east $=\hat x$, north $=\hat y$, up $=\hat z$ (so $\hat x\times\hat y=\hat z$), $\mathbf m=-m\hat y$ (south) and $\mathbf B=B\hat z$ (up): $$\boldsymbol\tau=\mathbf m\times\mathbf B=(-m\hat y)\times(B\hat z)=-mB(\hat y\times\hat z)=-mB\hat x=\boxed{0.010\ \text{N}\cdot\text{m, due west}}.$$ This is the sense that rotates the loop's south-pointing moment up toward alignment with $\mathbf B$, consistent with a torque that always acts to align $\mathbf m$ with $\mathbf B$.

Torque on the loop
Quantity	Result
Magnetic moment $m$	0.10 A·m² (pointing south)
Torque magnitude	0.010 N·m
Torque direction	due west

Topic: Torque on a current loop (magnetic dipole) in a uniform field.

Key relations: $\mathbf m=NI\mathbf A$ (right-hand rule sets direction); $\boldsymbol\tau=\mathbf m\times\mathbf B$; $\tau=mB\sin\theta$.

Why this works: a current loop's far-field behaviour is that of a magnetic dipole; the torque it feels in an external field always acts to rotate $\mathbf m$ toward alignment with $\mathbf B$, exactly analogous to an electric dipole in an electric field.

Common pitfall: applying the right-hand rule from the WRONG viewpoint — "clockwise viewed from due north" means the observer looks SOUTH at the loop, so the moment points away from that observer (south), not toward them (north). Getting this backwards flips the final torque direction (east instead of west).

Source: Hayt & Buck, Engineering Electromagnetics, Ch. 9 (magnetic forces, materials, and torque on a loop); Sadiku, Elements of Electromagnetics, Ch. 8.

Question 7 of 8: Torque on a Current Loop in a Uniform Field

Question 7: Torque on a Current Loop in a Uniform Field (equal value)