B Field Between Two Antiparallel Current Sheets: 04-BS-9, May 2015, Q4

Question 4: B Field Between Two Antiparallel Current Sheets (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
Sheet thickness $t$	$1$ mm $=10^{-3}$ m
Gap between sheets	$1$ mm
Current density $J$ (each sheet)	$2\ \text{A/mm}^2=2\times10^6\ \text{A/m}^2$
Directions	upper sheet: north; lower sheet: south (antiparallel)

Find. $\vec B$ in the gap between the sheets.

Upper sheet current north, lower sheet current south; the two sheets' contributions add in the gap (magnetic-capacitor analogy) and cancel outside.

Approach. A finite-thickness sheet carrying volume current density $J$ is equivalent, for field purposes, to a surface current $K=Jt$. Apply the standard infinite-sheet result $H=K/2$ on each side, then superpose both sheets: for antiparallel currents the two contributions add in the gap between them and cancel outside.

Equivalent surface current density per sheet. $$K=Jt=(2\times10^6)(10^{-3})$$ $$K=\boxed{2000\ \text{A/m}}$$
Superpose the two sheets in the gap. Each sheet contributes $K/2$ in the gap, and by the right-hand rule (current north on top, south on bottom) both contributions point the same way — west — so they add fully: $$H_{\text{gap}}=\frac{K}{2}+\frac{K}{2}=K=\boxed{2000\ \text{A/m}}$$ (outside the sheet pair the two contributions instead cancel, giving zero field — the magnetic analogue of a parallel-plate capacitor.)
Convert to flux density. $$B=\mu_0 H_{\text{gap}}=(4\pi\times10^{-7})(2000)$$ $$B=\boxed{2.513\times10^{-3}\ \text{T, directed West}}$$

Quantity	Result
$K$ (per sheet)	$2000$ A/m
$B$ in the gap	$2.513\times10^{-3}$ T, West

Concept & Theory

Topic. Magnetic field of finite-thickness antiparallel current sheets, the magnetic analogue of a parallel-plate capacitor.

Key relations. Volume-to-surface current conversion $K=Jt$; infinite-sheet field $H=K/2$ on each side, with sense fixed by $\hat n\times\vec K$ (right-hand rule).

Why this works. Each sheet, by symmetry, produces a uniform field of magnitude $K/2$ on either side, reversing sign across the sheet. Two sheets carrying opposite currents produce fields that point the same way in the region between them (adding to $K$) and opposite ways outside (cancelling to zero) — exactly analogous to how the E field of a parallel-plate capacitor adds between the plates and cancels outside.

Common pitfall. Treating the "$2\ \text{A/mm}^2$" as a surface current density directly (units already A/m, not A/m²) instead of a volume density that must be multiplied by the sheet thickness first.

Source. Sadiku, Elements of Electromagnetics, Ch. 7 (magnetic field of an infinite current sheet).

Question 4 of 8: B Field Between Two Antiparallel Current Sheets

Question 4: B Field Between Two Antiparallel Current Sheets (20 marks)