On-Axis B Field of a Horizontal Current Loop: 04-BS-9, May 2015, Q3

Question 3: On-Axis B Field of a Horizontal 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
Loop radius $a$	$1$ m
Current $I$	$1$ A
Field point height $z$	$1$ m above the centre, on the axis
Sense	clockwise, viewed from above

Find. $\vec B$ at the point on-axis, $z=1$ m above the centre.

Point P lies 1 m above the centre of a 1 m radius loop; clockwise current (viewed from above) gives B pointing down, toward the loop.

Approach. Apply the standard on-axis field of a circular loop, then fix the direction by the right-hand rule applied to the stated clockwise-from-above sense.

On-axis field magnitude. $$B=\frac{\mu_0 I a^2}{2(a^2+z^2)^{3/2}}=\frac{(4\pi\times10^{-7})(1)(1)^2}{2\left(1^2+1^2\right)^{3/2}}$$ $$B=\boxed{2.221\times10^{-7}\ \text{T}}$$
Direction. For a counter-clockwise current (viewed from above), the right-hand rule gives $\vec B$ pointing upward along $+z$ on the axis above the loop. The stated current is clockwise (the opposite sense), so $\vec B$ at the point above the loop points vertically downward, toward the loop.

Quantity	Result
$B$ at $z=1$ m	$2.221\times10^{-7}$ T, vertically downward

Concept & Theory

Topic. On-axis magnetic field of a circular current loop (Biot–Savart result).

Key relations. $B(z)=\mu_0Ia^2/2(a^2+z^2)^{3/2}$; at the centre ($z=0$) this reduces to $\mu_0I/2a$.

Why this works. Every current element contributes a $d\vec B$ with equal axial components and cancelling radial components by symmetry, so the on-axis field is purely along $z$ and given by this closed form; the sense follows directly from the right-hand rule applied to the loop's actual current direction.

Common pitfall. Using the centre-field formula ($z=0$) instead of the full on-axis expression, or getting the direction backward by applying the right-hand rule to a counter-clockwise assumption when the loop is explicitly clockwise.

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

Question 3 of 8: On-Axis B Field of a Horizontal Current Loop

Question 3: On-Axis B Field of a Horizontal Current Loop (20 marks)