Displacement Current – Definition, Maxwell’s Concept, Derivation, Equation, Physical Significance and Applications

Introduction

Displacement current is electromagnetic theory. It was introduced by the physicist James Clerk Maxwell to resolve a fundamental inconsistency in Ampere’s Circuital Law and to establish the complete set of Maxwell's equations. The concept of displacement current led to the prediction of electromagnetic waves and laid the foundation for modern communication technologies such as radio, television, radar, mobile phones, and wireless networks.

Unlike conduction current, which involves the actual movement of electric charges through a conductor, displacement current arises due to a changing electric field. Although no physical charge carriers move through an insulating medium, the changing electric field produces magnetic effects similar to those produced by ordinary electric current.

The introduction of displacement current unified electricity and magnetism into a single framework known as electromagnetism.

Historical Background

Before Maxwell's work, Ampere's Circuital Law was expressed as

∮H⋅dl=I

where:

• H = magnetic field intensity
• I = current enclosed by the path

This law worked perfectly for steady currents but failed in situations involving time-varying electric fields.

Maxwell observed a contradiction when applying Ampere's law to a charging capacitor. Current appeared to flow through the wires, but no conduction current existed between the capacitor plates. Yet a magnetic field was still present.

To solve this problem, Maxwell introduced the concept of displacement current.

What is Displacement Current?

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

It does not involve the actual flow of charge through a conductor but produces magnetic effects equivalent to those of conduction current.

Definition

Displacement current is defined as the rate of change of electric displacement flux through a surface.

Mathematically,

I_d​=d/dt ​∫_S​D⋅dS

where:

• I_d= displacement current
• D = electric flux density
• S = surface area

Maxwell's Modification of Ampere's Law

Maxwell proposed that the total current should consist of:

1. Conduction Current
2. Displacement Current

Thus,                       I_T​=I_c​+I_d​

The modified Ampere's law becomes

                             ∮H⋅dl=I_c​+I_d​

Or                      ∮H⋅dl=I_c​+d/dt ​∫_S​D⋅dS

This equation is called the Ampere-Maxwell Law.

Electric Flux Density

Electric flux density is defined as         

D= ε E

where:

• D = electric flux density
• E = electric field intensity
• ε = permittivity of the medium

Substituting into displacement current equation:

I_d​=ε d/dt​ ∫_S​E⋅dS

For uniform electric fields, 

  I_d​=ε  dΦ E​​/dt

where:                  Φ_E​=∫_S​E⋅dS

•  Φ_E is the electric flux.

Derivation of Displacement Current

Consider a parallel-plate capacitor connected to a battery.

When the switch is closed:

• Electrons move through the wire.
• Charges accumulate on capacitor plates.
• Electric field develops between the plates.

The conduction current in the wire is

                                   I_c=dQ/dt
where

Q is the charge on the plates.

The electric field between the plates is 

E= Q/ εA

where:

• A = plate area

Electric flux becomes  Φ_E​=EA

Substituting E,             Φ_E​ = Q​ /ε

Differentiating,          dΦ_E​​/dt=1/ ε dQ​/dt

Multiplying by ε,       dΦ_E​​/dt=dQ​/dt

Since                                       I_c​=dQ​/dt

therefore,                   I_d​=I_c​

Thus the displacement current between capacitor plates equals the conduction current in the external circuit.

Differential Form of Displacement Current

The displacement current density is   

J_d​= ∂D​/∂t

where:

• J_d= displacement current density

Using                           D=εE

                             J_d​=ε ∂E​/∂t

This shows that displacement current exists whenever the electric field changes with time.

Ampere-Maxwell Equation

In differential form,          

∇×H=J_c​+ ∂D​/∂t

where:

• J_c = conduction current density
• ∂D​/∂t  = displacement current density

This equation is one of Maxwell's four fundamental equations.

Physical Interpretation

Displacement current does not represent actual movement of charges through an insulating medium.

Instead, it represents:

• Time-varying electric fields.
• Continuity of current in a circuit.
• Magnetic field generation by changing electric fields.

Thus a changing electric field behaves like a current source.

Displacement Current in a Capacitor

The most common example occurs in a capacitor.

In the Wires

Current consists of moving electrons. 

I = I_c

Between Plates

No electrons cross the dielectric. 

I_c= 0

However, I_d ≠ 0

The changing electric field produces displacement current.

Hence        I_d=I_c
ensuring current continuity.

Need for Displacement Current

Without displacement current:

1. Ampere's law fails for capacitors.
2. Current continuity is violated.
3. Charge conservation appears inconsistent.
4. Electromagnetic wave theory becomes impossible.

Maxwell introduced displacement current to overcome these problems.

Relation with Continuity Equation

The continuity equation is 

∇⋅J=− ∂ρ​/∂t

where:

• J = current density
• ρ = charge density

Using Maxwell's correction,

∇⋅(J+ ∂D​/∂t)=0

Thus charge conservation is satisfied at every point.

Comparison Between Conduction Current and Displacement Current

Conduction Current	Displacement Current
Due to motion of charges	Due to changing electric field
Exists in conductors	Exists in dielectrics and free space
Requires charge carriers	No charge carriers required
Produces magnetic field	Produces magnetic field
Energy carried by moving electrons	Energy associated with electric field

Displacement Current in Free Space

One of Maxwell's greatest discoveries was that displacement current can exist even in vacuum.

For vacuum,           D=ε₀​E

Thus                        J_d​=ε₀ ∂E​/​∂t

A changing electric field in free space generates a magnetic field.

This idea led directly to electromagnetic wave propagation.

Role in Electromagnetic Waves

Maxwell showed:

• Changing electric field produces magnetic field.
• Changing magnetic field produces electric field.

According to     ∇×H=∂D​/∂t

And                    ∇×E=-∂B​/∂t

the two fields continuously regenerate each other.

This self-sustaining mechanism produces electromagnetic waves.

Examples include:

• Radio waves
• Microwaves
• Infrared radiation
• Visible light
• Ultraviolet radiation
• X-rays
• Gamma rays

Electromagnetic Wave Speed

Using Maxwell's equations,

c=1/√μ₀​ε₀

where:

•  c= speed of light
•  μ₀​= permeability of free space
• ε₀₌permittivity of free space

Maxwell calculated this value and found it equal to the measured speed of light.

This proved that light itself is an electromagnetic wave.

Significance of Displacement Current

Displacement current:

1. Completes Ampere's law.
2. Maintains current continuity.
3. Preserves charge conservation.
4. Explains capacitor operation.
5. Predicts electromagnetic waves.
6. Unifies electricity and magnetism.
7. Explains wave propagation in vacuum.

Applications of Displacement Current

1. Capacitors

Used in:

• Power supplies
• Filters
• Coupling circuits
• Tuning circuits

Displacement current exists in the dielectric region.

2. Radio Communication

Radio transmitters generate alternating electric fields.

These fields create displacement currents which produce electromagnetic waves.

3. Television Broadcasting

TV signals propagate through electromagnetic waves resulting from changing electric and magnetic fields.

4. Mobile Communication

Cellular networks rely on electromagnetic wave transmission.

Displacement current plays a key role in wave generation.

5. Radar Systems

Radar uses electromagnetic waves to detect objects.

Wave propagation depends on Maxwell's displacement current concept.

6. Satellite Communication

Signals travel through vacuum between satellites and Earth.

Displacement current enables electromagnetic wave transmission in free space.

7. Microwave Engineering

Microwave devices operate using rapidly varying electric fields that generate displacement currents.

8. Optical Fiber Communication

Light signals are electromagnetic waves whose existence is explained by displacement current theory.

Advantages of Maxwell's Concept

• Solves inconsistency in Ampere's law.
• Explains capacitor charging.
• Valid in vacuum and materials.
• Supports electromagnetic wave theory.
• Forms the basis of modern communication technology.

Limitations

• Not associated with actual charge transport.
• Difficult to measure directly.
• Significant mainly in time-varying fields.
• Negligible in many DC circuits.

Key Equations

• Displacement Current                     I_d​=d​/dt∫_S​D⋅dS
• Displacement Current Density      J_d​= ∂D​/∂t
• Ampere-Maxwell Law                  ∇×H=J_c​+ ∂D​/∂t
• Electric Flux Density                         D= εE
• Vacuum Form                                     J_d​=ε₀ ∂E​/​∂t

Conclusion

Displacement current is a revolutionary concept introduced by Maxwell to correct Ampere's Circuital Law and ensure the continuity of electric current in all situations. It arises from a changing electric field rather than the physical movement of electric charges. The concept explains how capacitors function, preserves charge conservation, and forms the basis of Maxwell's electromagnetic theory. Most importantly, displacement current led to the prediction and understanding of electromagnetic waves, making possible modern technologies such as radio communication, television, radar, satellites, mobile networks, and optical communication systems. Thus, displacement current remains one of the most significant ideas in classical electromagnetism and modern physics.