---
title: "Displacement current in parallel plate capacitor"
subject: "Engineering Physics"
chapter: 9
key: "semester_1/engineering_physics:displacement-current-circular-plates-b-integral"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/displacement-current-circular-plates-b-integral
site: BCT Engineering
---

# Displacement current in parallel plate capacitor

Engineering Physics · Chapter 9: Electromagnetic Waves

- Asked in: 2070 Asar, 5 marks
- Video: [A parallel plate capacitor made of circular plates is being charged such that the surface charge ...](https://www.youtube.com/watch?v=6-sRVT-g-rE) — Quantum Academy-By AB Sir (IISc AIR-1)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/displacement-current-circular-plates-b-integral
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9

## Question

A parallel plate capacitor with circular plates is charged by current 'i'. (a) What is the magnitude of ∮B·ds in terms of μ0 and i between the plates if r = (a/5) from the center? What is the magnitude of induced magnetic field for r = (a/5) in terms of displacement current?

## Short answer

∮B·ds = μ₀ i / 25. The induced magnetic field at r = a/5 is B = μ₀ i_d,enc/(2πr) = μ₀ i/(10πa), where i_d,enc = i/25.

## Answer

A parallel plate capacitor with circular plates of radius a is charged by a current i. Between the plates there is no conduction current, but the changing electric field gives a **displacement current** i_d = ε₀ dΦ_E/dt. The total displacement current through the whole plate area equals the charging current i. The displacement current density is uniform:

J_d = i/(πa²)

#### (a) Value of ∮B·ds at r = a/5
Ampere-Maxwell law: ∮B·ds = μ₀ i_d,enc, where i_d,enc is the displacement current inside the loop of radius r.

```text
i_d,enc = J_d × π r² = i × (r²/a²)
At r = a/5:  i_d,enc = i × (1/25) = i/25

∮B·ds = μ₀ i / 25
```

#### Induced magnetic field at r = a/5
```text
B (2π r) = μ₀ i_d,enc
B = μ₀ i_d,enc / (2π r) = μ₀ (i/25) / (2π a/5)
  = μ₀ i / (10 π a)
```

In general, inside the plates B = μ₀ i r/(2π a²), growing linearly with r, and outside B = μ₀ i/(2π r).

**Answer: ∮B·ds = μ₀i/25; B = μ₀ i_d,enc/(2πr) = μ₀i/(10πa), where i_d,enc = i/25.**

## Check yourself

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

1. What is the total displacement current through the whole plate area of a capacitor charged by current i?
   A. Zero
   B. i/2
   C. Equal to i
   D. i²
   Answer: C. The displacement current is continuous with the conduction current i. (Key point: Total displacement current)

2. At r = a/5, what fraction of the total displacement current is enclosed?
   A. 1/5
   B. 1/10
   C. 1/125
   D. 1/25
   Answer: D. Current is uniform over area, so the fraction is (r/a)² = 1/25. (Key point: Enclosed fraction)

3. What is ∮B·ds at r = a/5?
   A. μ₀ i/25
   B. μ₀ i/5
   C. μ₀ i
   D. μ₀ i/(10π a)
   Answer: A. ∮B·ds = μ₀ i_d,enc = μ₀ i/25. (Key point: Value of loop integral)

4. What is B at r = a/5?
   A. μ₀ i/(2π a)
   B. μ₀ i/(10π a)
   C. μ₀ i/(25 π a)
   D. μ₀ i/(50 π a)
   Answer: B. B = μ₀(i/25)/(2π a/5) = μ₀ i/(10π a). (Key point: Induced field)

## Related in Engineering Physics

- [Induced magnetic field, displacement current](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-induced-field-circular-plates) — Engineering Physics, chapter 9
- [Displacement current density, dE/dt, induced B](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-density-dedt-plates) — Engineering Physics, chapter 9
- [Displacement current: B at 50 mm](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-density-field-fifty-mm) — Engineering Physics, chapter 9
- [Parallel plate capacitor with mica](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amica-capacitor-field-free-charge) — Engineering Physics, chapter 7
- [Field at center of bent charged rod](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aplastic-rod-arc-field-center) — Engineering Physics, chapter 7
- [Circular coil field and magnetic dipole](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acircular-coil-axis-dipole) — Engineering Physics, chapter 8, asked 3×

## Related in other subjects

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