---
title: "Faraday's law and modified Maxwell equations"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:faraday-law-correct-curl-e-maxwell"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/faraday-law-correct-curl-e-maxwell
site: BCT Engineering
---

# Faraday's law and modified Maxwell equations

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2070 Chaitra, 1+3+4 marks
- Video: [FARADAY'S LAW AND MAXWELL'S EQUATION](https://www.youtube.com/watch?v=MDAWcRlcYAw) — BATHINA UPENDER RAO
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/faraday-law-correct-curl-e-maxwell
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

State Faraday's law and correct ∇×E = 0 for time varying field with necessary derivation. Also modify the equation ∇×H = J with necessary derivations for time varying field.

## Short answer

Faraday’s law: induced emf equals -dΨ/dt. For time-varying fields, ∇×E = -∂B/∂t. Ampere’s law ∇×H=J is inconsistent with charge conservation; adding displacement current ∂D/∂t gives ∇×H = J + ∂D/∂t.

## Answer

#### Faraday's law (1 mark)
The emf induced in a closed loop equals the negative rate of change of magnetic flux linking the loop:

emf = −dΨ/dt

The negative sign (Lenz's law) shows that the induced emf opposes the change producing it.

#### Correcting ∇×E = 0 (3 marks)
For static fields ∮ E·dl = 0, so ∇×E = 0. For a time-varying flux, the emf around a fixed loop is

```text
emf = ∮ E·dl = −dΨ/dt = −d/dt ∫ B·dS
```

For a stationary loop only B changes with time, so the derivative can go inside the integral:

```text
∮ E·dl = −∫ (∂B/∂t)·dS
```

Apply Stokes' theorem to the left side, ∮ E·dl = ∫ (∇×E)·dS:

```text
∫ (∇×E)·dS = −∫ (∂B/∂t)·dS
```

This holds for any surface, so the integrands are equal:

**∇×E = −∂B/∂t**

So the electric field is no longer conservative when B varies with time.

#### Modifying ∇×H = J (4 marks)
Ampere's law for steady currents is ∇×H = J. Take the divergence of both sides. Since ∇·(∇×H) = 0 always:

```text
∇·J = 0
```

But the continuity equation of charge says ∇·J = −∂ρv/∂t, which is non-zero for time-varying fields. So the equation is inconsistent. Add an unknown term J_x:

```text
∇×H = J + J_x   ⇒   ∇·J + ∇·J_x = 0   ⇒   ∇·J_x = ∂ρv/∂t
```

Using Gauss's law ρv = ∇·D:

```text
∇·J_x = ∂(∇·D)/∂t = ∇·(∂D/∂t)   ⇒   J_x = ∂D/∂t
```

Maxwell named this the displacement current density J_d = ∂D/∂t. The corrected equation is

**∇×H = J + ∂D/∂t**

and in integral form ∮ H·dl = ∫ (J + ∂D/∂t)·dS. This is consistent with charge conservation and predicts electromagnetic waves.

## Check yourself

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

1. What does the negative sign in emf = -d(psi)/dt represent?
   A. Energy is lost as heat
   B. The induced emf opposes the change producing it (Lenz's law)
   C. The flux is negative
   D. The loop is traversed clockwise
   Answer: B. The minus sign is Lenz's law: the induced emf opposes the change in flux. (Key point: Faraday's law)

2. What is the corrected form of curl E = 0 for time-varying fields?
   A. curl E = dB/dt
   B. curl E = -dD/dt
   C. curl E = J
   D. curl E = -dB/dt
   Answer: D. Equating the Stokes' theorem form with -integral(dB/dt . dS) for any surface gives curl E = -dB/dt. (Key point: Corrected curl E)

3. Why is curl H = J inconsistent for time-varying fields?
   A. div(curl H) = 0 forces div J = 0, contradicting the continuity equation
   B. J is always zero in time-varying fields
   C. H is not defined for varying fields
   D. It violates Gauss's law for B
   Answer: A. Taking the divergence gives div J = 0, but continuity requires div J = -d(rho_v)/dt, which is non-zero. (Key point: Why Ampere's law needs modifying)

4. Which added term fixes Ampere's law?
   A. dB/dt
   B. dD/dt
   C. dE/dx
   D. rho_v E
   Answer: B. Using rho_v = div D gives div J_x = div(dD/dt), so J_x = dD/dt and curl H = J + dD/dt. (Key point: Displacement current term)

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