---
title: "Displacement current: B at 50 mm"
subject: "Engineering Physics"
chapter: 9
key: "semester_1/engineering_physics:displacement-current-density-field-fifty-mm"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/displacement-current-density-field-fifty-mm
site: BCT Engineering
---

# Displacement current: B at 50 mm

Engineering Physics · Chapter 9: Electromagnetic Waves

- Asked in: 2067 Asoj, 5 marks
- Video: [Displacement current & Ampere Maxwell's law | Electromagnetic waves | Physics | Khan Academy](https://www.youtube.com/watch?v=t-W6_fn_1bI) — Khan Academy India - English
- 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-density-field-fifty-mm
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9

## Question

As a parallel plate capacitor with circular plates 20cm in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of 20A/m². Calculate the magnitude of magnetic field (B) at a distance r = 50mm from the axis of symmetry of this region. Also calculate dE/dt in this region.

## Short answer

At r = 50 mm, B = μ₀ J_d r/2 = 6.3×10⁻⁷ T. The rate of change of electric field is dE/dt = J_d/ε₀ = 2.3×10¹² V m⁻¹ s⁻¹.

## Answer

**Given:** plate diameter 20 cm, so R = 0.10 m; displacement current density J_d = 20 A/m² (uniform); r = 50 mm = 0.05 m (inside the plates since r < R).

#### Magnetic field at r = 50 mm
Ampere-Maxwell law (no conduction current between the plates): ∮B·dl = μ₀ I_d,enc.

```text
B (2π r) = μ₀ J_d (π r²)
B = μ₀ J_d r / 2
  = (4π×10⁻⁷ × 20 × 0.05) / 2
  = 6.28×10⁻⁷ T
```

#### dE/dt
J_d = ε₀ dE/dt, so

```text
dE/dt = J_d / ε₀ = 20 / 8.85×10⁻¹²
      = 2.26×10¹² V/(m·s)
```

**Answer: B ≈ 6.3×10⁻⁷ T; dE/dt ≈ 2.3×10¹² V m⁻¹s⁻¹**

## Check yourself

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

1. The plate radius is 0.10 m and r = 50 mm. Which formula applies?
   A. Inside the plates, B = μ₀ J_d r/2
   B. Outside, B = μ₀ I/(2π r)
   C. B = 0 because r < R
   D. B = μ₀ J_d R²/(2r)
   Answer: A. Since r < R, only the displacement current within radius r is enclosed. (Key point: Which region)

2. What is B at r = 0.05 m for J_d = 20 A/m²?
   A. About 1.26×10⁻⁶ T
   B. About 6.3×10⁻⁷ T
   C. About 6.3×10⁻⁴ T
   D. About 2.5×10⁻⁶ T
   Answer: B. B = 4π×10⁻⁷ × 20 × 0.05/2 = 6.28×10⁻⁷ T. (Key point: Computing B)

3. How is dE/dt found?
   A. dE/dt = J_d ε₀
   B. dE/dt = μ₀ J_d
   C. dE/dt = J_d/ε₀ ≈ 2.3×10¹² V/(m·s)
   D. dE/dt = J_d/μ₀
   Answer: C. Since J_d = ε₀ dE/dt. (Key point: dE/dt)

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