---
title: "Maxwell's equations, displacement current"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:maxwell-equations-explain-displacement-depth"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/maxwell-equations-explain-displacement-depth
site: BCT Engineering
---

# Maxwell's equations, displacement current

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2076 Asoj, 10 marks
- Video: [Maxwell's Equations: Crash Course Physics #37](https://www.youtube.com/watch?v=K40lNL3KsJ4) — CrashCourse
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/maxwell-equations-explain-displacement-depth
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

State and explain Maxwell's equations in differential and integral form. Also define displacement current and depth of penetration.

## Short answer

Maxwell's equations: div D = rho_v, integral D·dS = Q; div B = 0, integral B·dS = 0; curl E = -dB/dt, integral E·dl = -d/dt integral B·dS; curl H = J + dD/dt, integral H·dl = integral (J + dD/dt)·dS. Displacement current J_d = dD/dt ensures continuity. Depth delta = sqrt(2/(omega mu sigma)), distance where field falls to 1/e.

## Answer

#### Maxwell's equations
Maxwell's four equations are the complete set of laws of classical electromagnetism. They combine Gauss's law (electric), Gauss's law (magnetic), Faraday's law and Ampere's circuital law (corrected with displacement current).

| Law | Differential (point) form | Integral form |
|---|---|---|
| Gauss's law (electric) | ∇·D = ρv | ∮ D·dS = ∫ ρv dv = Q |
| Gauss's law (magnetic) | ∇·B = 0 | ∮ B·dS = 0 |
| Faraday's law | ∇×E = −∂B/∂t | ∮ E·dl = −d/dt ∫ B·dS |
| Ampere's law | ∇×H = J + ∂D/∂t | ∮ H·dl = ∫ (J + ∂D/∂t)·dS |

Explanation of each:
1. **Electric Gauss law:** electric flux leaving a closed surface equals the free charge enclosed. Charge is the source of D.
2. **Magnetic Gauss law:** there are no magnetic monopoles. Magnetic flux lines are closed loops, so net B flux through any closed surface is zero.
3. **Faraday's law:** a time-varying magnetic flux induces an emf (a circulating E field). The minus sign is Lenz's law.
4. **Ampere's law:** a magnetic field circulates around conduction current J and around a changing electric flux (displacement current ∂D/∂t).

Supporting relations: D = εE, B = μH, J = σE. For a static field, the time derivatives vanish and the equations reduce to the static laws.

#### Displacement current
Ampere's law in the form ∇×H = J fails for time-varying fields because ∇·(∇×H) = 0 would force ∇·J = 0, which contradicts the continuity equation ∇·J = −∂ρv/∂t. Maxwell added the term

**J_d = ∂D/∂t (A/m²)**

so that ∇×H = J + J_d. Displacement current is the current "through" a dielectric or vacuum due to a changing electric flux density. It is not a flow of charges, but it produces a magnetic field like a real current. It makes current continuous through a capacitor and is the reason electromagnetic waves can exist in free space.

#### Depth of penetration (skin depth)
When a wave enters a conducting medium, its amplitude decays as e^(−αz). The depth of penetration δ is the distance at which the amplitude falls to 1/e (about 36.8 %) of its surface value:

δ = 1/α

For a good conductor (σ ≫ ωε), α = β = √(πfμσ), so

δ = √(2/(ωμσ)) = 1/√(πfμσ)

A higher frequency or higher conductivity gives a smaller δ, so the fields and currents stay near the surface (skin effect). Example: copper at 1 MHz has δ ≈ 66 μm.

## Check yourself

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

1. Which is the point form of Faraday's law?
   A. ∇×E = −∂B/∂t
   B. ∇×H = J + ∂D/∂t
   C. ∇·B = 0
   D. ∇·D = ρ_v
   Answer: A. A changing magnetic flux induces a circulating E; the minus sign is Lenz's law. (Key point: Faraday's law)

2. ∇·B = 0 means:
   A. No magnetic monopoles, flux lines are closed loops
   B. B is zero
   C. B is uniform
   D. Charges produce B
   Answer: A. Net B flux through any closed surface is zero. (Key point: Magnetic Gauss law)

3. Displacement current J_d = ∂D/∂t:
   A. Is a flow of free charges
   B. Is due to changing flux and creates a magnetic field like real current
   C. Exists only in conductors
   D. Is zero in vacuum
   Answer: B. It is not charge flow but produces H, completes current through capacitors, and allows EM waves. (Key point: Displacement current)

4. The depth of penetration δ is where amplitude falls to:
   A. 1/e of surface value, δ = 1/α
   B. 50%
   C. 1/e², δ = 2/α
   D. Zero
   Answer: A. Amplitude decays as e^(−αz); at z = 1/α it is 36.8%. Higher f or σ gives smaller δ. (Key point: Skin depth)

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