---
title: "Frequency where conduction equals displacement"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:displacement-current-conduction-ratio-soil"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-conduction-ratio-soil
site: BCT Engineering
---

# Frequency where conduction equals displacement

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2069 Chaitra, 2+5 marks
- Video: [Conduction and Displacement current densities/Problem/Unit_4/EMT](https://www.youtube.com/watch?v=ersEP9cAIRg) — Dr.Jayaudhaya ,Simple and Easy Way
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-conduction-ratio-soil
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Define displacement current. Assume that dry soil has conductivity equal to 10⁻⁴ S/m, ε = 3ε0 and μ = μ0. Determine the frequency at which the ratio of the magnitudes of conduction current density to displacement current density is unity.

## Short answer

Displacement current density is J_d = ∂D/∂t = ε ∂E/∂t, introduced by Maxwell to make Ampere’s law consistent. For unity ratio |J_c|/|J_d| = σ/(ωε) = 1, solve f = σ/(2πε) = 10⁻⁴/(2π·3·8.854×10⁻¹²) ≈ 5.99×10⁵ Hz (≈600 kHz).

## Answer

#### Displacement current (2 marks)
Displacement current is the current that exists in a dielectric or free space due to a time-varying electric flux density. Its density is J_d = ∂D/∂t = ε ∂E/∂t (A/m²). It was introduced by Maxwell into Ampere's law, ∇×H = J + ∂D/∂t, to make it consistent with the continuity of charge. It produces a magnetic field like a conduction current.

#### Numerical (5 marks)
Take E = E_m cos ωt (any time-harmonic field). Then:

```text
J_c = σ E             → magnitude σ E_m
J_d = ε ∂E/∂t         → magnitude ω ε E_m
```

Ratio of magnitudes (this is also the loss tangent):

```text
|J_c| / |J_d| = σ / (ω ε)
```

For the ratio to be unity: σ = ω ε = 2π f ε, so

```text
f = σ / (2π ε)
```

**Given:** σ = 10⁻⁴ S/m, ε = 3ε₀ = 3 × 8.854×10⁻¹² = 2.656×10⁻¹¹ F/m.

```text
f = 10⁻⁴ / (2π × 2.656×10⁻¹¹)
  = 10⁻⁴ / (1.669×10⁻¹⁰)
  = 5.99×10⁵ Hz
```

**Answer: f ≈ 599 kHz (about 0.6 MHz).** Below this frequency, conduction current dominates (soil behaves like a conductor); above it, displacement current dominates (soil behaves like a dielectric).

## Check yourself

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

1. Displacement current density is:
   A. J_d = sigma E
   B. J_d = dD/dt = epsilon dE/dt
   C. J_d = rho_v v
   D. J_d = dB/dt
   Answer: B. It arises from a time-varying electric flux density and was added by Maxwell to Ampere's law. (Key point: Definition of displacement current)

2. What is |J_c|/|J_d| for time-harmonic fields?
   A. w epsilon / sigma
   B. sigma epsilon / w
   C. sigma / (w epsilon)
   D. sigma w epsilon
   Answer: C. J_c has magnitude sigma E_m and J_d has magnitude w epsilon E_m, so the ratio is sigma/(w epsilon), the loss tangent. (Key point: Ratio of currents)

3. For the ratio to be 1, the frequency is:
   A. f = sigma epsilon / 2 pi
   B. f = 2 pi epsilon / sigma
   C. f = epsilon / (2 pi sigma)
   D. f = sigma / (2 pi epsilon)
   Answer: D. Setting sigma = 2 pi f epsilon gives f = sigma/(2 pi epsilon). (Key point: Frequency for unity ratio)

4. For sigma = 10^-4 S/m and epsilon = 3 epsilon0, f is about:
   A. 599 kHz
   B. 59.9 kHz
   C. 5.99 MHz
   D. 599 Hz
   Answer: A. f = 10^-4/(2 pi x 2.656 x 10^-11) = 5.99 x 10^5 Hz. (Key point: Numerical result)

## Related in Electromagnetism

- [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×
- [Conduction and displacement current in capacitor](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aconduction-displacement-current-capacitor) — Electromagnetism, chapter 3
- [Displacement current density in transformer](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-density-transformer) — Electromagnetism, chapter 3
- [Displacement current density at 60 Hz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-metallic-conductor-60hz) — Electromagnetism, chapter 3
- [Maxwell's equations, displacement current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-explain-displacement-depth) — Electromagnetism, chapter 3

## Related in other subjects

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- [Theoretical maximum dry density](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Atheoretical-maximum-dry-density) — Soil Mechanics, chapter 5
- [Conductivity of sodium from electron density](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asodium-conductivity-electron-density) — Engineering Physics, chapter 8, asked 2×
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