---
title: "Maxwell's equations and continuity equation"
subject: "Engineering Physics"
chapter: 9
key: "semester_1/engineering_physics:maxwell-continuity-equation"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/maxwell-continuity-equation
site: BCT Engineering
---

# Maxwell's equations and continuity equation

Engineering Physics · Chapter 9: Electromagnetic Waves

- Asked in: 2069 Asar, 5 marks; 2069 Chaitra, 5 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/maxwell-continuity-equation
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9

## Question

State and explain Maxwell's equations (in free space and dielectric medium). With the help of Maxwell's equations, derive the continuity equation (charge conservation theorem) ∇·J = −∂ρ/∂t.

## Short answer

Take divergence of ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. Using ∇·(∇×B)=0 and Gauss's law ∇·E=ρ/ε₀ gives ∇·J = −∂ρ/∂t, the continuity equation expressing charge conservation.

## Answer

#### Maxwell's equations
**In free space** (ρ = 0, J = 0):

| Law | Differential form |
|---|---|
| Gauss (electric) | ∇·E = 0 |
| Gauss (magnetic) | ∇·B = 0 |
| Faraday | ∇×E = −∂B/∂t |
| Ampere-Maxwell | ∇×B = μ₀ε₀ ∂E/∂t |

**In a dielectric medium** with free charge density ρ and free current density J, using D = εE and H = B/μ:

| Law | Differential form |
|---|---|
| Gauss (electric) | ∇·D = ρ |
| Gauss (magnetic) | ∇·B = 0 |
| Faraday | ∇×E = −∂B/∂t |
| Ampere-Maxwell | ∇×H = J + ∂D/∂t |

Meaning: ∇·D = ρ says charges are the source of D. ∇·B = 0 says there are no monopoles. Faraday's law says a changing B produces E. The last says currents and a changing D (displacement current) produce H.

#### Continuity equation
1. Start with the Ampere-Maxwell law in vacuum with sources: ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.
2. Take the divergence of both sides. The divergence of a curl is always zero:

   0 = μ₀ ∇·J + μ₀ε₀ ∂(∇·E)/∂t
3. Use Gauss's law ∇·E = ρ/ε₀:

   0 = μ₀ ∇·J + μ₀ε₀ ∂(ρ/ε₀)/∂t = μ₀ ∇·J + μ₀ ∂ρ/∂t
4. Divide by μ₀:

**∇·J = −∂ρ/∂t**

This says the net current flowing out of a small volume equals the rate of decrease of charge inside it, which is conservation of charge. Without Maxwell's displacement term, step 2 would give ∇·J = 0, which is wrong for charging capacitors.

## Check yourself

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

1. Which equation is the starting point for deriving ∇·J = −∂ρ/∂t?
   A. The Ampere-Maxwell law ∇×B = μ₀J + μ₀ε₀ ∂E/∂t
   B. Faraday's law ∇×E = −∂B/∂t
   C. ∇·B = 0 alone
   D. The Biot-Savart law
   Answer: A. Its divergence, together with Gauss's law, produces the continuity equation. (Key point: Starting point for continuity)

2. Why does taking the divergence of the Ampere-Maxwell law help?
   A. The divergence of J is always zero
   B. The divergence of a curl is always zero, so the remaining terms sum to zero
   C. It removes ε₀ from the equation
   D. It converts B into E
   Answer: B. ∇·(∇×B) = 0 forces μ₀∇·J + μ₀ε₀ ∂(∇·E)/∂t = 0. (Key point: Why divergence of curl matters)

3. In a dielectric medium with free charge ρ and free current J, Gauss's electric law and the Ampere-Maxwell law are written as:
   A. ∇·E = 0 and ∇×H = J
   B. ∇·D = 0 and ∇×H = ∂D/∂t only
   C. ∇·D = ρ and ∇×H = J + ∂D/∂t
   D. ∇·B = ρ and ∇×E = J
   Answer: C. Using D and H handles the medium, with free sources ρ and J. (Key point: Dielectric form)

4. What would the divergence of Ampere's law give without Maxwell's displacement term?
   A. ∇·J = −∂ρ/∂t
   B. ∇·J = ρ
   C. ∇·J = ∂B/∂t
   D. ∇·J = 0, which is wrong for a charging capacitor
   Answer: D. Without the term, charge could not build up on capacitor plates, contradicting charge conservation. (Key point: Role of displacement term)

## Related in Engineering Physics

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- [Maxwell's equations, integral to differential](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-integral-to-differential-explain) — Engineering Physics, chapter 9, asked 3×
- [EM wave equations in vacuum](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-differential-free-space-wave-equations) — Engineering Physics, chapter 9, asked 2×
- [Capacitance of dielectric-filled spherical shells](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aspherical-shells-dielectric-capacitance) — Engineering Physics, chapter 7

## Related in other subjects

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- [Derive differential continuity equation](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Acontinuity-equation-differential-form) — Applied Computational Fluid Dynamics, chapter 2