---
title: "Maxwell's equations, integral to differential"
subject: "Engineering Physics"
chapter: 9
key: "semester_1/engineering_physics:maxwell-integral-to-differential-explain"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/maxwell-integral-to-differential-explain
site: BCT Engineering
---

# Maxwell's equations, integral to differential

Engineering Physics · Chapter 9: Electromagnetic Waves

- Asked in: 2072 Kartik, 5 marks; 2070 Asar, 5 marks; 2067 Asoj, 5 marks (asked 3 times)
- Video: [Maxwell's equations from integral to differential form | Lecture 53 | Vector Calculus for Engineers](https://www.youtube.com/watch?v=l5YF42lmdBI) — Jeffrey Chasnov
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/maxwell-integral-to-differential-explain
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9

## Question

State Maxwell's equations in integral form. Convert them into differential form. Explain each of these equations.

## Short answer

Integral form: ∮E·dA = q/ε₀, ∮B·dA = 0, ∮E·dl = −dΦ_B/dt, ∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt. Differential form: ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. They describe electric and magnetic field sources, induction, and conservation.

## Answer

#### Maxwell's equations in integral form
1. ∮E·dA = q/ε₀ (Gauss's law for electricity)
2. ∮B·dA = 0 (Gauss's law for magnetism)
3. ∮E·dl = −dΦ_B/dt (Faraday's law)
4. ∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt (Ampere-Maxwell law)

#### Conversion to differential form
We use the divergence theorem (∮F·dA = ∫∇·F dV) and Stokes' theorem (∮F·dl = ∫(∇×F)·dA), with q = ∫ρ dV and I = ∫J·dA.

1. ∫∇·E dV = (1/ε₀)∫ρ dV for any volume, so **∇·E = ρ/ε₀**.
2. ∫∇·B dV = 0 for any volume, so **∇·B = 0**.
3. ∫(∇×E)·dA = −∫(∂B/∂t)·dA for any surface, so **∇×E = −∂B/∂t**.
4. ∫(∇×B)·dA = μ₀∫J·dA + μ₀ε₀∫(∂E/∂t)·dA, so **∇×B = μ₀J + μ₀ε₀ ∂E/∂t**.

#### Meaning of each equation
| Equation | Meaning |
|---|---|
| ∇·E = ρ/ε₀ | Electric charges are the sources of E; field lines start on + and end on − charge. |
| ∇·B = 0 | No magnetic monopoles; B lines are closed loops. |
| ∇×E = −∂B/∂t | A changing magnetic field produces a circulating electric field (induction). |
| ∇×B = μ₀J + μ₀ε₀∂E/∂t | Currents and changing electric fields produce a circulating magnetic field. |

The last term, introduced by Maxwell (displacement current), makes charge conservation consistent and predicts electromagnetic waves. In free space ρ = 0 and J = 0.

## Check yourself

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

1. Which theorem converts the integral form of Gauss's laws into differential form?
   A. Divergence theorem
   B. Stokes' theorem
   C. Green's reciprocity theorem
   D. Gauss-Bolyai theorem
   Answer: A. The closed-surface integral of F·dA equals the volume integral of div F, which gives ∇·E and ∇·B. (Key point: Tool for surface integral to divergence)

2. Applying Stokes' theorem to ∮E·dl = −dΦ_B/dt gives which differential equation?
   A. ∇·E = −∂B/∂t
   B. ∇×E = −∂B/∂t
   C. ∇×E = +∂B/∂t
   D. ∇×B = −∂E/∂t
   Answer: B. Stokes' theorem turns the line integral into the surface integral of curl E, matching −∂B/∂t. (Key point: Faraday's law differential form)

3. What does ∇·B = 0 tell us?
   A. Magnetic field is always zero
   B. Changing E produces B
   C. There are no magnetic monopoles and B lines are closed loops
   D. Magnetic field lines start on north poles and end at infinity
   Answer: C. Zero divergence means B has no sources or sinks, so its lines close on themselves. (Key point: Meaning of ∇·B = 0)

4. What is the role of the term μ₀ε₀ ∂E/∂t in the Ampere-Maxwell law?
   A. It represents conduction current in a wire
   B. It represents magnetic monopole current
   C. It is zero in all situations
   D. It is the displacement current, making charge conservation consistent and predicting EM waves
   Answer: D. Maxwell's displacement term lets a changing E produce B and leads to electromagnetic waves. (Key point: Role of Maxwell's added term)

## Related in Engineering Physics

- [EM wave in dielectric from Maxwell's equations](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-em-wave-dielectric-velocity) — Engineering Physics, chapter 9, asked 2×
- [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×
- [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×

## Related in other subjects

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- [Maxwell's equations in phasor form](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-phasor-free-space) — Electromagnetism, chapter 3, asked 2×
- [Maxwell's equations, displacement current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-explain-displacement-depth) — Electromagnetism, chapter 3
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