---
title: "Ampere's law vs continuity equation"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:ampere-law-conflict-continuity"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/ampere-law-conflict-continuity
site: BCT Engineering
---

# Ampere's law vs continuity equation

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2074 Asoj, 4 marks; 2072 Chaitra, 2+3 marks (asked 2 times)
- Video: [Differential Form of Faraday's and Ampere's law, Equation of Continuity, Limitation of Ampere's law](https://www.youtube.com/watch?v=-5EV_ezDNi4) — RKag
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/ampere-law-conflict-continuity
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

How does ∇×H = J conflict with the continuity equation in time varying fields? How is this conflict rectified (corrected) in such fields?

## Short answer

Static Ampere's law ∇×H = J implies ∇·J = 0, contradicting continuity ∇·J = −∂ρ_v/∂t for time-varying fields. Maxwell corrected it by adding displacement current ∂D/∂t: ∇×H = J + ∂D/∂t, making it consistent.

## Answer

#### The conflict
Ampere's law for static fields is ∇×H = J. Take the divergence of both sides. The divergence of a curl is always zero:

```text
∇·(∇×H) = 0 = ∇·J
```
So Ampere's law demands ∇·J = 0. But the continuity equation (conservation of charge) states

```text
∇·J = −∂ρ_v/∂t
```
For time-varying fields ∂ρ_v/∂t ≠ 0 (for example, charging a capacitor), so ∇·J ≠ 0. The two equations contradict each other. Ampere's law is valid only for steady (DC) currents.

#### The correction (Maxwell's displacement current)
Add an unknown term J_d to the right side: ∇×H = J + J_d. Then ∇·J + ∇·J_d = 0, so ∇·J_d = −∇·J = ∂ρ_v/∂t. Using Gauss's law ρ_v = ∇·D:

```text
∇·J_d = ∂(∇·D)/∂t = ∇·(∂D/∂t)    →    J_d = ∂D/∂t
```
The corrected equation is

```text
∇×H = J + ∂D/∂t
```
Now ∇·(J + ∂D/∂t) = 0, which agrees with continuity. The term ∂D/∂t is the **displacement current density** (A/m²). It flows in the dielectric between capacitor plates and completes the current loop, even though no real charges move across the gap. In integral form: ∮H·dl = I_conduction + I_displacement.

## Check yourself

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

1. Why does ∇×H = J conflict with continuity?
   A. ∇·(∇×H) = 0 forces ∇·J = 0, but continuity gives ∇·J = −∂ρ_v/∂t
   B. Curl H is always infinite
   C. J is a scalar
   D. Continuity applies only to static fields
   Answer: A. For time-varying charge density ∂ρ_v/∂t ≠ 0, so the two cannot both hold. (Key point: Source of conflict)

2. How did Maxwell fix Ampere's law?
   A. Removed J
   B. Added displacement current: ∇×H = J + ∂D/∂t
   C. Added ∂B/∂t
   D. Replaced H with B
   Answer: B. ∇·J_d = ∂ρ_v/∂t with ρ_v = ∇·D gives J_d = ∂D/∂t. (Key point: Fix)

3. After correction, ∇·(J + ∂D/∂t) equals:
   A. ρ_v
   B. ∂ρ_v/∂t
   C. 0
   D. −J
   Answer: C. It equals the divergence of a curl, which is zero, consistent with continuity. (Key point: Check)

## Related in Electromagnetism

- [Fields from time varying B](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Afields-from-time-varying-b) — Electromagnetism, chapter 3
- [State Ampere's circuital law](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aampere-law-statement-with-example) — Electromagnetism, chapter 3, asked 2×
- [Ampere's law and boundary H components](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aamperes-law-surface-current-boundary) — Electromagnetism, chapter 3
- [Relaxation time constant from continuity](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Arelaxation-time-continuity-derive) — Electromagnetism, chapter 2, asked 2×
- [Derive continuity equation](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acontinuity-equation-derive-forms) — Electromagnetism, chapter 2, asked 2×
- [Faraday's law and modified Maxwell equations](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Afaraday-law-correct-curl-e-maxwell) — Electromagnetism, chapter 3

## Related in other subjects

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- [Ampere's law, Maxwell's modification](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aampere-law-maxwell-modification-displacement) — Engineering Physics, chapter 9
- [Ampere's law, field of straight wire](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aampere-law-straight-wire-field) — Engineering Physics, chapter 8, asked 5×
- [Maxwell's equations and continuity equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-continuity-equation) — Engineering Physics, chapter 9, asked 2×