Displacement Current

In classical electromagnetic theory, Ampere's law was originally stated only for steady currents flowing through conductors. However, this formulation failed to account for situations where electric fields change with time, such as inside a charging capacitor. James Clerk Maxwell identified this inconsistency and introduced the concept of displacement current to complete the theory, making it consistent with the continuity equation and enabling the prediction of electromagnetic waves.

Core Concept Explanation

Consider a parallel plate capacitor being charged by an alternating current source. Conduction current flows through the connecting wires, but there are no free charges crossing the gap between the plates. If you apply Ampere's original law to a closed loop around the wire, you get a nonzero result. If you apply it to a surface that passes through the capacitor gap (instead of cutting through the wire), you get zero. This contradiction violates the self-consistency of Maxwell's equations.

Maxwell resolved this by adding a new term to Ampere's law. He noticed that the electric field between the capacitor plates changes with time as charge accumulates, and that the electric flux through the gap is proportional to the charge on the plates. The rate of change of this flux is equal in magnitude to the conduction current in the wire. Maxwell called this rate of change of electric flux the displacement current, denoted Jd, even though no physical charge actually moves through the gap.

The displacement current density is defined as the time derivative of the electric flux density D. Since D equals the product of permittivity and electric field intensity, the displacement current density becomes the product of permittivity and the time rate of change of E. This term has the same units as conduction current density (A/m²) and plays the same mathematical role in generating a magnetic field.

Mathematical Expression

The displacement current density is expressed as:

Jd = dD/dt = ε · dE/dt

The total current density in Maxwell's corrected Ampere's law then becomes the sum of conduction current density and displacement current density:

∇ × H = J + dD/dt

In integral form, the total current enclosed by a closed path equals the sum of conduction current and the rate of change of electric flux through the bounded surface:

∮ H · dl = I_conduction + d/dt ∫∫ D · dS

For a linear isotropic medium with permittivity ε, D equals εE. In free space, ε equals ε₀ (8.854 × 10⁻¹² F/m). In a medium with relative permittivity εr, ε equals ε₀εr. The displacement current contributes to the magnetic field exactly like conduction current does, which is why the two are interchangeable in closed-loop analysis.

Practical Understanding

The physical interpretation of displacement current is tied to the concept of electric flux. When the electric field between capacitor plates increases, the electric flux through any cross-section of the gap increases. This time-varying flux is mathematically equivalent to a current threading that surface. Even though no charge physically moves through the dielectric, the changing field produces a magnetic field around the gap just as a wire current would.

This concept is fundamental to understanding electromagnetic wave propagation. In a wave traveling through free space, neither conduction current nor free charges are present. The oscillating electric field produces a displacement current, which generates a magnetic field. That magnetic field in turn generates another electric field via Faraday's law. This self-sustaining cycle is the physical basis of electromagnetic wave propagation.

In practical engineering, displacement current becomes significant at high frequencies. At low frequencies (like 50 Hz power lines), displacement current is negligible compared to conduction current. At microwave and RF frequencies, displacement current inside dielectric materials and across capacitor gaps becomes the primary mechanism of signal transmission.

Given:
Parallel plate capacitor with plate area A = 4 cm² = 4×10⁻⁴ m²
Separation d = 2 mm (gap filled with air, ε₀ = 8.854×10⁻¹² F/m)
Applied voltage V(t) = 100 sin(2π×10⁶ t) volts

Why this formula applies:
The electric field between plates is E = V/d, and displacement current density Jd = ε₀ · dE/dt.

Formula:
Jd = ε₀ · dE/dt = (ε₀/d) · dV/dt
Id = Jd × A

Substitution:
dV/dt = 100 × 2π × 10⁶ × cos(2π×10⁶ t)  [maximum = 100 × 2π × 10⁶]
Jd_max = (8.854×10⁻¹²) / (2×10⁻³) × (100 × 2π × 10⁶)
Jd_max = 4.427×10⁻⁹ × 6.2832×10⁸

Calculation:
Jd_max = 4.427×10⁻⁹ × 6.2832×10⁸ = 2.782 A/m²
Id_max = 2.782 × 4×10⁻⁴

Final Answer with units:
Id_max ≈ 1.113 mA (peak displacement current through the capacitor gap)

Exam Tip: In GATE problems, displacement current in a capacitor equals conduction current in the connecting wire at every instant. Use Id = C · dV/dt as a shortcut when capacitance is given directly. Do not confuse displacement current density Jd (A/m²) with total displacement current Id (A).

• Displacement current arises due to the time-varying electric flux density D between capacitor plates, not due to physical movement of charges.
• Its density is Jd = dD/dt = ε · dE/dt, which has the same units as conduction current density (A/m²).
• Displacement current equals conduction current in the connecting wire at every instant, ensuring current continuity around the entire circuit loop.
• In free space and dielectric media with no free charges, displacement current is the only source term for the magnetic field.
• Displacement current is the fundamental mechanism behind electromagnetic wave propagation in vacuum and dielectric materials.

Quick Revision

• Displacement current density: Jd = dD/dt = ε · dE/dt (units: A/m²).
• Maxwell's corrected Ampere's law: ∇ × H = J + dD/dt.
• Inside a charging capacitor: displacement current = conduction current in wire at every moment.
• Displacement current is not actual charge flow; it is the mathematical effect of changing electric flux.
• At high frequencies, displacement current becomes dominant over conduction current in dielectric regions.
• Shortcut formula: Id = C · dV/dt when capacitance is known directly.
• Exam trap: Displacement current does not flow through conductors; it exists only in regions where E varies with time and no free charges are present.