Displacement Current | Maxwell's Modification of Ampere's Law

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1. Introduction

Ampere's circuital law relates the magnetic field around a closed path to the electric current enclosed by that path.

For a steady current, Ampere's law is written as:

∮ B · dl = μ₀ I

However, this form creates a problem when the current is changing with time, especially during the charging of a capacitor.

Maxwell solved this problem by introducing the concept of displacement current.

2. The Problem with Ampere's Law

Consider a capacitor being charged by a battery. Current flows through the connecting wires, but there is no ordinary conduction current through the insulating gap between the capacitor plates.

🔋 Battery

→

⚡ Conduction Current

→

🔵 Capacitor

The electric field between the capacitor plates changes with time. This changing electric field produces a magnetic field.

3. 🔄 Charging Capacitor Animation

Observe the changing electric field between the capacitor plates. This changing field is associated with displacement current.

BATTERY

↑
⚡
↓

⬆

Observation: There is no conduction current across the capacitor gap, but the changing electric field behaves like a current in producing a magnetic field.

4. What is Displacement Current?

Displacement current is the current associated with a time-varying electric field.

It is not an actual flow of electric charges through the dielectric gap like conduction current.

I<sub>d</sub> = ε₀ dΦ<sub>E</sub>/dt

Where:

I<sub>d</sub>
Displacement current

ε₀
Permittivity of free space

Φ<sub>E</sub>
Electric flux

dΦ<sub>E</sub>/dt
Rate of change of electric flux

5. Maxwell's Modification of Ampere's Law

Maxwell added the displacement current term to Ampere's law.

∮ B · dl = μ₀(I + I<sub>d</sub>)

Therefore:

∮ B · dl = μ₀I + μ₀ε₀ dΦ<sub>E</sub>/dt

6. General Maxwell-Ampere Law

∇ × B = μ₀J + μ₀ε₀ ∂E/∂t

This equation shows that magnetic fields can be produced by:

🔌 Conduction Current

Produced by actual movement of electric charges.

⚡ Changing Electric Field

Produces displacement current and therefore contributes to the magnetic field.

7. Displacement Current in a Charging Capacitor

For a parallel plate capacitor:

E = Q / (ε₀A)

Electric flux is:

Φ<sub>E</sub> = EA

Therefore:

Φ<sub>E</sub> = Q / ε₀

Differentiating with respect to time:

dΦ<sub>E</sub>/dt = 1/ε₀ × dQ/dt

Since:

dQ/dt = I

We obtain:

I<sub>d</sub> = I

Important Result:

During charging of an ideal capacitor, the displacement current between the plates is equal in magnitude to the conduction current in the wires.

8. Conduction Current vs Displacement Current

Property	Conduction Current	Displacement Current
Origin	Movement of charges	Changing electric field
Symbol	I	Id
Expression	I = dQ/dt	Id = ε₀dΦE/dt
Capacitor gap	No conduction current	Displacement current exists
Produces magnetic field?	Yes	Yes

9. ⭐ Key Concept

Changing Electric Field

→

Displacement Current

→

Magnetic Field

The major contribution of Maxwell was recognizing that a time-varying electric field can produce a magnetic field.

10. 📐 Important Formulae

I<sub>d</sub> = ε₀ dΦ<sub>E</sub>/dt

∮ B · dl = μ₀(I + I<sub>d</sub>)

I<sub>d</sub> = I
For an ideal charging capacitor

∇ × B = μ₀J + μ₀ε₀ ∂E/∂t

11. 🧮 Worked Example

The electric flux through a surface changes at a rate of 2 × 10<sup>6</sup> N m²/C per second. Find the displacement current.

I<sub>d</sub> = ε₀ dΦ<sub>E</sub>/dt

ε₀ = 8.85 × 10<sup>−12</sup> C²/(N m²)

I<sub>d</sub> = (8.85 × 10<sup>−12</sup>) (2 × 10<sup>6</sup>)

I<sub>d</sub> = 1.77 × 10<sup>−5</sup> A

Answer: 1.77 × 10<sup>−5</sup> A

12. 🎯 Practice MCQs

Q1. Maxwell introduced displacement current to modify:

A. Faraday's law B. Ampere's law C. Ohm's law D. Coulomb's law

✅ Correct Answer: B. Ampere's law

Q2. Displacement current is associated with:

A. Constant electric field B. Time-varying electric field C. Constant magnetic field D. Static charge only

✅ Correct Answer: B. Time-varying electric field

Q3. The expression for displacement current is:

A. I = V/R B. I = dQ/dt C. I<sub>d</sub> = ε₀dΦ<sub>E</sub>/dt D. I = QR

✅ Correct Answer: C. I<sub>d</sub> = ε₀dΦ<sub>E</sub>/dt

Q4. During charging of an ideal capacitor:

A. I<sub>d</sub> = 0 B. I<sub>d</sub> = I C. I<sub>d</sub> = 2I D. I<sub>d</sub> = I/2

✅ Correct Answer: B. I<sub>d</sub> = I

Q5. A changing electric field produces:

A. Only electric charge B. Magnetic field C. Zero field D. Only heat

✅ Correct Answer: B. Magnetic field

13. 🏆 One-Minute Revision

Ampere's Law
Relates magnetic field to current.

Maxwell's Modification
Added displacement current.

Displacement Current
I<sub>d</sub> = ε₀dΦ<sub>E</sub>/dt

Charging Capacitor
I<sub>d</sub> = I

⭐ Remember

Changing Electric Field → Displacement Current → Magnetic Field

⚡ Displacement Current | Maxwell's Modification of Ampere's Law
Interactive Physics Learning • JEE / CBSE

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