Magnetic Flux Density Midway Between Two Parallel Current Loops: 04-BS-9, May 2016, Q3

Question 3: Magnetic Flux Density Midway Between Two Parallel Current Loops (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 $a$	$5$ cm $=0.05$ m
Loop-to-loop separation	$2.5$ cm $=0.025$ m
Current $I$ (each loop, same sense)	$0.1$ A
Field point	midpoint of the common axis, $x=1.25$ cm from each loop

Find. The magnitude and direction of $\vec B$ at the centre of the system (the axial midpoint).

Two coaxial loops (5 cm radius, 2.5 cm apart) carrying same-sense current; both contribute equally to $B$ at the axial midpoint.

Approach. Use the on-axis field of a single circular loop for each loop at its own distance from the midpoint ($x=1.25$ cm), then add the two contributions — they reinforce because the currents circulate the same way.

On-axis field of one loop at $x=1.25$ cm from its plane. $$B_{\text{one}}=\frac{\mu_0 I a^2}{2(a^2+x^2)^{3/2}}=\frac{(4\pi\times10^{-7})(0.1)(0.05)^2}{2\left[(0.05)^2+(0.0125)^2\right]^{3/2}}$$ $$B_{\text{one}}=\boxed{1.147\times10^{-6}\ \text{T}}$$
Add the two loops' contributions. Both loops sit the same distance ($1.25$ cm) from the midpoint and circulate the same way, so their axial fields point the same way and add directly: $$B_{\text{net}}=2\,B_{\text{one}}=\boxed{2.295\times10^{-6}\ \text{T, along the common axis (sense set by the right-hand rule for the stated current direction)}}$$

Quantity	Result
Each loop's contribution $B_{\text{one}}$	$1.147\times10^{-6}$ T
$B_{\text{net}}$ at the centre	$2.295\times10^{-6}$ T, along the common axis

Concept & Theory

Topic. On-axis magnetic field of circular current loops (Biot–Savart), superposed for a two-loop system.

Key relations. $B(x)=\dfrac{\mu_0 I a^2}{2(a^2+x^2)^{3/2}}$ on the axis of a single loop of radius $a$, current $I$, at distance $x$ from its plane.

Why this works. Each loop's field at any axial point is purely axial by symmetry, so the two loops' contributions at the shared midpoint are collinear and add or subtract as scalars depending on whether the currents circulate the same way (add, as here) or oppositely (subtract) — this two-loop geometry is the same "aid" configuration used for Helmholtz coils, just at a spacing other than the Helmholtz-optimal $a$.

Common pitfall. Using the small-$x$ (near-field, $x\ll a$) approximation $B\approx\mu_0 I/(2a)$ instead of the full $(a^2+x^2)^{3/2}$ denominator; here $x=1.25$ cm is not negligible next to $a=5$ cm ($x/a=0.25$), so the full expression is needed for four-significant-figure accuracy.

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

Question 3 of 8: Magnetic Flux Density Midway Between Two Parallel Current Loops

Question 3: Magnetic Flux Density Midway Between Two Parallel Current Loops (20 marks)