---
title: "Displacement current density in transformer"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:displacement-current-density-transformer"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-density-transformer
site: BCT Engineering
---

# Displacement current density in transformer

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2075 Chaitra, 6 marks
- Video: [Displacement Current | Questions 1 | Electromagnetic Theory](https://www.youtube.com/watch?v=iwMknJ-3Cts) — ENGINEERING TUTORIAL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-density-transformer
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Find the amplitude of the displacement current density in air within a large power transformer where H = 10⁶ cos(377t + 1.2566×10⁻⁶z) a_y A/m.

## Short answer

In air, displacement current density J_d = curl H. With H = 10^6 cos(377t + 1.2566e-6 z) a_y A/m, curl H = -dH_y/dz a_x = 1.2566 sin(377t + 1.2566e-6 z) a_x A/m^2. Amplitude is 1.26 A/m^2.

## Answer

In air σ = 0, so there is no conduction current (J = 0). From Maxwell's equation ∇×H = J + ∂D/∂t, the displacement current density equals the curl of H:

J_d = ∂D/∂t = ∇×H

**Given:** H = H_y a_y with H_y = 10⁶ cos(377t + 1.2566×10⁻⁶ z) A/m. H depends only on z.

**Step 1: curl of H.**

```text
∇×H = (∂H_z/∂y − ∂H_y/∂z) a_x + (∂H_x/∂z − ∂H_z/∂x) a_y + (∂H_y/∂x − ∂H_x/∂y) a_z
```

Only H_y(z) exists, so only the a_x component survives:

```text
∇×H = −∂H_y/∂z a_x
    = −10⁶ × (−sin(377t + 1.2566×10⁻⁶ z)) × 1.2566×10⁻⁶ a_x
    = 1.2566 sin(377t + 1.2566×10⁻⁶ z) a_x   A/m²
```

**Step 2: interpretation.** In air the conduction part is zero, so the whole of ∇×H is displacement current. Since sin θ = cos(θ − 90°), J_d lags H by 90° in time.

**Answer: J_d = 1.2566 sin(377t + 1.2566×10⁻⁶ z) a_x A/m², so the amplitude is 1.2566 A/m²** (about 1.26 A/m²).

## Check yourself

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

1. In air (σ = 0), the displacement current density equals:
   A. ∇×H
   B. ∇·H
   C. σE
   D. ∂B/∂t
   Answer: A. ∇×H = J + ∂D/∂t with J = 0 gives J_d = ∇×H. (Key point: Relation)

2. With H = H_y(z) a_y, ∇×H has component:
   A. −∂H_y/∂z a_x
   B. ∂H_y/∂x a_z
   C. ∂H_y/∂z a_y
   D. 0
   Answer: A. Only the x-component survives. (Key point: Surviving component)

3. The amplitude of J_d is:
   A. 1.2566 A/m²
   B. 10^6 A/m²
   C. 1.2566×10^−6 A/m²
   D. 377 A/m²
   Answer: A. 10^6 × 1.2566×10^−6 = 1.2566 A/m², with a sine time dependence. (Key point: Amplitude)

## Related in Electromagnetism

- [Displacement current density at 60 Hz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-metallic-conductor-60hz) — Electromagnetism, chapter 3
- [Displacement current density at 1 kHz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-metal-1khz) — Electromagnetism, chapter 3
- [Displacement vs conduction current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-vs-conduction-current) — Electromagnetism, chapter 3, asked 2×
- [Maxwell's equations, displacement current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-explain-displacement-depth) — Electromagnetism, chapter 3
- [Conduction and displacement current in capacitor](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aconduction-displacement-current-capacitor) — Electromagnetism, chapter 3
- [Frequency where conduction equals displacement](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-conduction-ratio-soil) — Electromagnetism, chapter 3

## Related in other subjects

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- [Air gap flux density in core with three windings](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Athree-winding-core-airgap-flux-density) — Electrical Machines I, chapter 1
- [No load current of single phase transformer](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Atransformer-no-load-current) — Electric Machine Design, chapter 3
- [Air gap and rotor bars, 10 kW motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Acage-rotor-10kw-design-data) — Electric Machine Design, chapter 4