---
title: "Total current through z = 4 surface"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:current-through-z-4-surface"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/current-through-z-4-surface
site: BCT Engineering
---

# Total current through z = 4 surface

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2069 Chaitra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/current-through-z-4-surface
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

The magnetic field intensity is given in a certain region of space as H = [(x + 2y)/z²] a_y + (2/z) a_z A/m. Find the total current passing through the surface z = 4, 1 < x < 5, 3 < y < 5, in the a_z direction.

## Short answer

For surface z=4, I = ∫ (∇×H)_z dS. (∇×H)_z = ∂/∂x[(x+2y)/z²] = 1/z². Integrating over 1≤x≤5, 3≤y≤5 at z=4 gives I = (1/16)×4×2 = 0.5 A.

## Answer

The total current through a surface is I = ∫ J·dS, with J = ∇×H (Ampere's law, static or conduction form).

H = [(x + 2y)/z²] a_y + (2/z) a_z, so H_x = 0, H_y = (x + 2y)/z², H_z = 2/z.

**Step 1: a_z component of the curl** (the surface z = 4 has normal a_z).

```text
(∇×H)_z = ∂H_y/∂x − ∂H_x/∂y
        = ∂/∂x [(x + 2y)/z²] − 0
        = 1/z²
```

(For information, the other components are (∇×H)_x = ∂H_z/∂y − ∂H_y/∂z = 2(x+2y)/z³ and (∇×H)_y = 0, but they do not cross the plane z = 4.)

**Step 2: integrate over the surface** z = 4, 1 < x < 5, 3 < y < 5, dS = dx dy a_z.

```text
I = ∫_{y=3}^{5} ∫_{x=1}^{5} (1/z²) dx dy   at z = 4
  = (1/16) × (5 − 1) × (5 − 3)
  = 8/16
  = 0.5 A
```

**Check with Stokes' theorem:** go round the rectangle at z = 4 (counter-clockwise seen from +z). H_x = 0, so only the edges x = 5 (going +y) and x = 1 (going −y) contribute.

```text
x = 5 edge: ∫₃⁵ (5 + 2y)/16 dy = 26/16 = 1.625
x = 1 edge: ∫₃⁵ (1 + 2y)/16 dy = 18/16 = 1.125
I = 1.625 − 1.125 = 0.5 A
```

This matches.

**Answer: I = 0.5 A** in the a_z direction.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The surface is z = 4. Which component of curl H gives the current in a_z?
   A. (curl H)_x
   B. (curl H)_y
   C. (curl H)_z = dHy/dx - dHx/dy
   D. The magnitude of curl H
   Answer: C. The normal is a_z, so only (curl H)_z = dH_y/dx - dH_x/dy contributes. (Key point: Component of curl needed)

2. For H_y = (x+2y)/z^2 and H_x = 0, what is (curl H)_z?
   A. 2/z^2
   B. 1/z^2
   C. (x+2y)/z^3
   D. 0
   Answer: B. d/dx[(x+2y)/z^2] = 1/z^2; the 2y term does not contribute to the x-derivative. (Key point: Value of (curl H)_z)

3. What is the total current?
   A. 0.25 A
   B. 1 A
   C. 0.5 A
   D. 2 A
   Answer: C. I = (1/16) x (5-1) x (5-3) = 8/16 = 0.5 A. (Key point: Integration)

4. In the Stokes' theorem check, which edges contribute?
   A. Only the edges x = 1 and x = 5 (H_x = 0 so the y-edges give nothing)
   B. All four edges equally
   C. Only the y = 3 and y = 5 edges
   D. None, because H_z is constant
   Answer: A. H_x = 0, so edges running along x give nothing; the x = 5 edge (1.625) minus the x = 1 edge (1.125) gives 0.5 A. (Key point: Stokes check)

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