Magnetic Field Between Three Coaxial Current-Carrying Cylinders: 04-BS-9, May 2017, Q2

Question 2: Magnetic Field Between Three Coaxial Current-Carrying Cylinders (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.

Cylinder radius	Current
$r_1=1$mm	$1$A, direction "$-$"
$r_2=2$mm	$2$A, direction "$+$" (opposite sense)
$r_3=3$mm	$1$A, direction "$-$"

Find. The minimum and maximum $B$ in each of the two gaps: $r_1<r<r_2$ and $r_2<r<r_3$.

Three coaxial thin cylindrical current sheets (cross-section); the two annular gaps are $1$mm–$2$mm and $2$mm–$3$mm.

Approach. Ampère's law with a circular loop of radius $r$: $B(r)=\mu_0I_{\text{enc}}/(2\pi r)$, where $I_{\text{enc}}$ is the (signed) sum of currents inside radius $r$. Within a gap, $I_{\text{enc}}$ is constant, so $B\propto1/r$ — extremes occur at the gap's two edges.

Gap 1 ($1$mm$<r<2$mm): only cylinder 1 is enclosed. $I_{\text{enc}}=1$A (magnitude), constant across the gap: $$B(r_1^+)=\frac{\mu_0(1)}{2\pi(0.001)}=\boxed{2.000\times10^{-4}\ \text{T (max, at }r=1\text{mm)}}$$ $$B(r_2^-)=\frac{\mu_0(1)}{2\pi(0.002)}=\boxed{1.000\times10^{-4}\ \text{T (min, at }r=2\text{mm)}}$$
Gap 2 ($2$mm$<r<3$mm): cylinders 1 and 2 enclosed. Opposite directions partially cancel: $I_{\text{enc}}=|-1+2|=1$A, constant across the gap: $$B(r_2^+)=\frac{\mu_0(1)}{2\pi(0.002)}=\boxed{1.000\times10^{-4}\ \text{T (max, at }r=2\text{mm)}}$$ $$B(r_3^-)=\frac{\mu_0(1)}{2\pi(0.003)}=\boxed{6.667\times10^{-5}\ \text{T (min, at }r=3\text{mm)}}$$

Region	Minimum $B$	Maximum $B$
Gap 1 ($1$mm–$2$mm)	$1.000\times10^{-4}$ T (at $r=2$mm)	$2.000\times10^{-4}$ T (at $r=1$mm)
Gap 2 ($2$mm–$3$mm)	$6.667\times10^{-5}$ T (at $r=3$mm)	$1.000\times10^{-4}$ T (at $r=2$mm)

Concept & Theory

Topic. Ampère's law for coaxial infinitesimally-thin current sheets — a stepwise-constant enclosed current gives a piecewise $1/r$ field with jumps only AT each sheet.

Key relations. $B(r)=\mu_0I_{\text{enc}}(r)/(2\pi r)$; $I_{\text{enc}}$ changes only when $r$ crosses a current sheet, and is constant (hence $B\propto1/r$, monotonic) within each gap.

Why this works. Because a thin cylindrical current sheet contributes to $I_{\text{enc}}$ as an all-or-nothing step function of $r$ (unlike Question 4's uniform-density conductor, where the enclosed fraction grows continuously), the field in each source-free annular gap is exactly the single-wire $1/r$ law with whatever net current happens to be enclosed there — so the extremes are always at the gap's inner and outer edges, never in the interior.

Common pitfall. Forgetting to net the OPPOSING directions when tallying $I_{\text{enc}}$ in gap 2 (using $1+2=3$A instead of the signed sum $|-1+2|=1$A), or assuming $B$ is constant across a gap rather than $1/r$-varying.

Source. Sadiku, Elements of Electromagnetics, Ch. 7 (Ampère's circuital law, coaxial conductors).

Question 2 of 8: Magnetic Field Between Three Coaxial Current-Carrying Cylinders

Question 2: Magnetic Field Between Three Coaxial Current-Carrying Cylinders (20 marks)