---
title: "All Maxwell equations, point and integral"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:maxwell-equations-point-integral-form"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/maxwell-equations-point-integral-form
site: BCT Engineering
---

# All Maxwell equations, point and integral

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2070 Chaitra, 4 marks
- Video: [Maxwell Equations Explained: Integral Form and Differential Form/Point Form](https://www.youtube.com/watch?v=dk2WEWsOEHo) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/maxwell-equations-point-integral-form
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Write all the Maxwell equations for the time varying field in point form as well as integral form.

## Short answer

Maxwell's equations: ∇·D=ρ_v, ∮D·dS=∫ρ_v dv; ∇·B=0, ∮B·dS=0; ∇×E=-∂B/∂t, ∮E·dl=-∫∂B/∂t·dS; ∇×H=J+∂D/∂t, ∮H·dl=∫(J+∂D/∂t)·dS.

## Answer

The four Maxwell's equations for time-varying fields (with ρv the free volume charge density, J the conduction current density):

| No. | Law | Point (differential) form | Integral form |
|---|---|---|---|
| 1 | Gauss's law for E | ∇·D = ρv | ∮ D·dS = ∫ ρv dv |
| 2 | Gauss's law for B | ∇·B = 0 | ∮ B·dS = 0 |
| 3 | Faraday's law | ∇×E = −∂B/∂t | ∮ E·dl = −∫ (∂B/∂t)·dS |
| 4 | Ampere's law (Maxwell's form) | ∇×H = J + ∂D/∂t | ∮ H·dl = ∫ (J + ∂D/∂t)·dS |

Meaning:
- Equation 1: electric flux out of a closed surface = enclosed charge.
- Equation 2: no isolated magnetic charge; B flux lines are closed.
- Equation 3: a changing magnetic field produces an electric field.
- Equation 4: a current or a changing electric field produces a magnetic field.

Constitutive relations: D = εE, B = μH, J = σE. The continuity equation ∇·J = −∂ρv/∂t follows from equations 1 and 4.

## Check yourself

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

1. Which is Maxwell's Faraday equation in point form?
   A. curl E = dB/dt
   B. curl E = -dB/dt
   C. div E = -dB/dt
   D. curl B = -dE/dt
   Answer: B. A changing magnetic field produces an electric field: curl E = -dB/dt. (Key point: Faraday's law form)

2. Which equation includes the displacement current?
   A. div D = rho_v
   B. div B = 0
   C. curl H = J + dD/dt
   D. curl E = -dB/dt
   Answer: C. Maxwell's form of Ampere's law adds dD/dt to J. (Key point: Ampere-Maxwell law)

3. What does div B = 0 (closed-surface integral of B . dS = 0) express?
   A. Electric charges are conserved
   B. Magnetic field is uniform
   C. There are no isolated magnetic charges; B flux lines are closed
   D. B is always zero
   Answer: C. Net B flux through any closed surface is zero because no magnetic monopoles exist. (Key point: Gauss's law for B)

4. Which is the integral form of Gauss's law for E?
   A. Closed integral of D . dS = integral of rho_v dv
   B. Closed integral of E . dl = 0
   C. Closed integral of B . dS = integral of rho_v dv
   D. Closed integral of D . dl = Q
   Answer: A. Electric flux out of a closed surface equals the enclosed charge. (Key point: Gauss's law for D)

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