H Field Between Two Antiparallel Infinite Current Sheets: 04-BS-9, December 2013, Q3

Question 3: H Field Between Two Antiparallel Infinite 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 $t$ (each)	$1$ mm $=1\times10^{-3}$ m
Gap between sheets	$1$ mm
Volume current density $J$ (each sheet)	$10^{-3}$ A/m²
Current directions	upper sheet North, lower sheet South (antiparallel)

Find. The magnitude and direction of $\vec H$ in the gap between the two sheets.

Two infinite current sheets 1 mm apart, currents antiparallel (North / South). Fields add between the sheets and cancel outside.

Approach. Reduce each finite-thickness sheet to an equivalent infinitesimally thin surface current $K=Jt$, apply the standard result $\vec H=\tfrac12\vec K\times\hat a_N$ for one infinite sheet, then superpose the two sheets' contributions in the gap.

Equivalent surface current density. Each sheet's current per unit width, collapsed to a sheet of zero thickness: $$K=Jt=(10^{-3}\ \text{A/m}^2)(1\times10^{-3}\ \text{m})$$ $$K=\boxed{1\times10^{-6}\ \text{A/m}}$$
Field of one infinite sheet. For a sheet with surface current $\vec K$, $\vec H=\tfrac12\vec K\times\hat a_N$, where $\hat a_N$ is the unit normal from the sheet toward the field point. Take East$=\hat x$, North$=\hat y$, Up$=\hat z$. The gap lies below the upper (North-flowing) sheet, so $\hat a_N=-\hat z$ there: $$\vec H_{\text{upper}}=\tfrac12(K\hat y)\times(-\hat z)=-\tfrac{K}{2}\hat x$$ The gap lies above the lower (South-flowing) sheet, so $\hat a_N=+\hat z$ there: $$\vec H_{\text{lower}}=\tfrac12(-K\hat y)\times(\hat z)=-\tfrac{K}{2}\hat x$$
Superpose. Both contributions point the same way in the gap and add (this is the sheet-pair/"magnetic capacitor" configuration — fields add between the sheets, cancel outside): $$\vec H_{\text{gap}}=\vec H_{\text{upper}}+\vec H_{\text{lower}}=-K\hat x$$ $$\vec H_{\text{gap}}=\boxed{1\times10^{-6}\ \text{A/m, directed West}}$$

Quantity	Result
Equivalent surface current $K$	$1\times10^{-6}$ A/m
$\vec H$ between the sheets	$1\times10^{-6}$ A/m, West

Concept & Theory

Topic. Ampere's law for infinite current sheets; a sheet pair as a "magnetic-field capacitor."

Key relations. $\vec H=\tfrac12\vec K\times\hat a_N$ for one infinite sheet; for two antiparallel sheets, $H=K$ between them and $H=0$ outside — the magnetic analogue of the field between (and outside) a parallel-plate capacitor.

Why this works. Each sheet alone produces a uniform, oppositely-directed field on its two sides. Between two antiparallel sheets, both sheets' fields point the same way and add; outside the pair, the two sheets' fields point opposite ways and cancel — exactly analogous to how two oppositely-charged sheets confine $\vec E$ to the gap.

Common pitfall. Treating the given $10^{-3}$ A/m² as if it were already a surface current density (A/m) — the sheets have explicit thickness, so it must first be multiplied by that thickness to get $K$.

Source. Sadiku, Elements of Electromagnetics, Ch. 7 (Ampere's Circuit Law, magnetic field of an infinite sheet of current).

Question 3 of 8: H Field Between Two Antiparallel Infinite Current Sheets

Question 3: H Field Between Two Antiparallel Infinite Current Sheets (20 marks)