On-Axis Field of an Atomic-Scale Current Loop: 04-BS-9, December 2016, Q5

Question 5: On-Axis Field of an Atomic-Scale Current Loop (20 marks)

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.

Quantity	Value
Current $I$	$10$ mA $=0.01$ A
Loop radius $a$	$10^{-10}$ m
Field point height $z$	$10^{-10}$ m above centre (so $z=a$)
Current sense (viewed from above)	clockwise

Find. The magnitude and direction of $\vec B$ at the field point.

Loop of radius $a$ carries current $I$ circulating clockwise viewed from above; the field point sits on the axis at height $z=a$.

Approach. Apply the standard on-axis field of a circular loop with $z=a$; determine the SIGN by the right-hand rule applied to the stated (clockwise-from-above) current sense.

Magnitude, on-axis loop field with $z=a$. $$B=\frac{\mu_0Ia^2}{2(a^2+z^2)^{3/2}}=\frac{\mu_0Ia^2}{2(2a^2)^{3/2}}=\frac{\mu_0I}{4\sqrt2\,a}$$ $$B=\frac{(4\pi\times10^{-7})(0.01)}{4\sqrt2\,(10^{-10})}=\boxed{22.21\ \text{T}}$$
Direction by the right-hand rule. Curling the right-hand fingers in the current's sense as seen from above (clockwise) points the thumb DOWNWARD, i.e. the loop's magnetic moment $\vec m$ points down through the loop; the on-axis field is directed along $\vec m$ at every point on the axis (both above and below the loop, not just far away): $$\boxed{\vec B=22.21\ \text{T, directed DOWNWARD (toward the loop), along the axis}}$$

Quantity	Result
Magnitude $B$	$22.21$ T
Direction	downward along the axis (toward the loop), set by the clockwise-from-above current

Check: a loop radius and field height of $10^{-10}$ m (atomic scale) with a macroscopic $10$ mA current gives an enormous field ($22.2$ T, stronger than any practical laboratory magnet); this is a direct, expected consequence of the given atomic-scale geometry (consistent with Questions 1, 2, and 7 on this same paper, which are also set at Ångstrom scale), not an arithmetic error.

Concept & Theory

Topic. On-axis magnetic field of a circular current loop (Biot–Savart), and the right-hand-rule sign convention linking current sense to field direction.

Key relations. $B(z)=\mu_0Ia^2/[2(a^2+z^2)^{3/2}]$; at $z=a$ this simplifies to $\mu_0I/(4\sqrt2\,a)$.

Why this works. The on-axis field of ANY planar current loop points along the loop's magnetic moment $\vec m=I\vec A$ for every point on the axis, not merely in the far-field limit; a clockwise sense viewed from above corresponds to $\vec m$ pointing down (opposite to the more commonly assumed counter-clockwise/upward convention), which is exactly why the problem specifies the viewing sense explicitly.

Common pitfall. Defaulting to the "upward" moment convention (counter-clockwise from above) without checking the problem's stated sense, which would flip the reported direction (though not the magnitude).

Source. Sadiku, Elements of Electromagnetics, Ch. 7 (Biot–Savart law, on-axis field of a circular loop, right-hand rule).

Question 5 of 8: On-Axis Field of an Atomic-Scale Current Loop

Question 5: On-Axis Field of an Atomic-Scale Current Loop (20 marks)