1.5 Displacement current

Displacement current concept

Displacement current fills a gap in classical electromagnetism: it extends the idea of electric current to include time-varying electric fields, not just moving charges. Without it, Ampère's original law breaks down in situations like a charging capacitor, where current flows in the wires but no charge crosses the gap between the plates. Maxwell's fix resolved this inconsistency and, as a bonus, predicted electromagnetic waves.

Capacitor charging and current

When a capacitor charges, current flows through the external wires, yet no charge physically crosses the gap between the plates. So what sustains the magnetic field in that region? The answer is the changing electric field between the plates, which acts as a source of magnetic field just like a real current would. This "current" due to the changing field is the displacement current, and it's proportional to the rate of change of the electric field in the gap.

Electric field and displacement current

The displacement current density is defined as:

$\vec{J}_D = \frac{\partial \vec{D}}{\partial t}$

where $\vec{D}$ is the electric displacement field. In a linear medium, $\vec{D}$ relates to the electric field through the permittivity:

$\vec{D} = \varepsilon \vec{E}$

So in free space, $\vec{J}_D = \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$. Anywhere the electric field changes with time, a displacement current density exists, whether or not charges are present.

Displacement current vs. conduction current

• Conduction current involves actual charge carriers moving through a conductor under the influence of an electric field.
• Displacement current involves no moving charges at all. It arises purely from a time-varying electric field.
• In any closed loop, the sum of conduction current and displacement current is continuous. This is what guarantees charge conservation across boundaries like capacitor gaps.

Displacement current equation

Displacement current formula

The displacement current density is:

$\vec{J}_D = \frac{\partial \vec{D}}{\partial t}$

To get the total displacement current $I_D$ through a surface $S$, integrate the density over that surface:

$I_D = \int_S \vec{J}_D \cdot d\vec{A} = \int_S \frac{\partial \vec{D}}{\partial t} \cdot d\vec{A}$

In free space, where $\vec{D} = \varepsilon_0 \vec{E}$, this simplifies to:

$I_D = \varepsilon_0 \frac{d\Phi_E}{dt}$

where $\Phi_E = \int_S \vec{E} \cdot d\vec{A}$ is the electric flux through the surface.

Permittivity and electric flux

Permittivity $\varepsilon$ quantifies how a medium responds to an applied electric field. It connects $\vec{E}$ and $\vec{D}$:

$\vec{D} = \varepsilon \vec{E}$

The electric flux through a surface is:

$\Phi_E = \int_S \vec{E} \cdot d\vec{A}$

Note that some texts define electric flux using $\vec{D}$ instead of $\vec{E}$. Be careful about which convention your course uses, since it affects where factors of $\varepsilon_0$ appear.

Displacement current derivation

Here's the core argument for why displacement current must exist:

1. Start with Ampère's original law in differential form: $\nabla \times \vec{B} = \mu_0 \vec{J}$
2. Take the divergence of both sides: $\nabla \cdot (\nabla \times \vec{B}) = \mu_0 \nabla \cdot \vec{J}$
3. The left side is identically zero (divergence of any curl vanishes), so $\nabla \cdot \vec{J} = 0$.
4. But the continuity equation requires $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$, which is not zero when charge density is changing (e.g., on capacitor plates).
5. To fix this, add a term: $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$
6. Now taking the divergence and using Gauss's law ($\nabla \cdot \vec{E} = \rho / \varepsilon_0$) gives $\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0$, which is exactly the continuity equation.

The added term $\mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ is the displacement current contribution. It's not optional; without it, Maxwell's equations are internally inconsistent.

Displacement current applications

Displacement current in capacitors

Consider a parallel-plate capacitor with plate area $A$ being charged by a current $I$. Between the plates, the electric field is $E = \sigma / \varepsilon_0 = Q / (\varepsilon_0 A)$, so:

$I_D = \varepsilon_0 \frac{d\Phi_E}{dt} = \varepsilon_0 A \frac{dE}{dt} = \frac{dQ}{dt} = I$

The displacement current between the plates exactly equals the conduction current in the wires. This ensures that the magnetic field is consistent regardless of which surface you choose for your Amperian loop, whether it passes through the wire or through the gap.

Displacement current in vacuum

Displacement current doesn't require matter. In vacuum, any region where $\vec{E}$ changes with time has a displacement current density $\varepsilon_0 \frac{\partial \vec{E}}{\partial t}$. This is the insight that led Maxwell to predict electromagnetic waves: a changing electric field creates a magnetic field, which in turn creates a changing electric field, and so on, sustaining a wave that propagates through empty space.

Displacement current in dielectrics

In a dielectric, the displacement field $\vec{D}$ accounts for both the free-space contribution and the polarization of bound charges:

$\vec{D} = \varepsilon_0 \vec{E} + \vec{P}$

where $\vec{P}$ is the polarization. The displacement current density $\frac{\partial \vec{D}}{\partial t}$ therefore includes contributions from the changing polarization as well. For a linear dielectric with permittivity $\varepsilon = \varepsilon_0 \varepsilon_r$, this becomes $\varepsilon \frac{\partial \vec{E}}{\partial t}$, which is larger than the vacuum value by a factor of $\varepsilon_r$.

Ampère's circuital law

Ampère's law without displacement current

The original form of Ampère's law relates the circulation of $\vec{B}$ around a closed loop to the enclosed conduction current:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

This works perfectly for magnetostatics, where currents are steady and fields don't change with time. The trouble appears as soon as fields vary.

Inconsistency with the continuity equation

The continuity equation expresses conservation of charge:

$\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$

As shown in the derivation above, taking the divergence of the original Ampère's law forces $\nabla \cdot \vec{J} = 0$, which contradicts the continuity equation whenever charge is accumulating or depleting somewhere. A charging capacitor is the classic example: charge builds up on the plates, so $\frac{\partial \rho}{\partial t} \neq 0$, and the original law fails.

Generalized Ampère's law with displacement current

Maxwell's correction adds the displacement current term:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)$

In differential form:

$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

This is the Ampère-Maxwell law. It's fully consistent with charge conservation and reduces to the original Ampère's law in the static limit where $\frac{\partial \vec{E}}{\partial t} = 0$.

Maxwell's equations

The four Maxwell equations, taken together, completely describe classical electrodynamics. With the displacement current included, they form a self-consistent set.

Gauss's law for electric fields

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \qquad \text{or} \qquad \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$

Electric flux through a closed surface is determined by the enclosed charge.

Gauss's law for magnetic fields

$\oint \vec{B} \cdot d\vec{A} = 0 \qquad \text{or} \qquad \nabla \cdot \vec{B} = 0$

There are no magnetic monopoles; magnetic field lines always close on themselves.

Faraday's law of induction

$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \qquad \text{or} \qquad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

A changing magnetic flux induces an electric field. The negative sign reflects Lenz's law.

Ampère-Maxwell law

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt} \qquad \text{or} \qquad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

Magnetic fields are generated by both conduction currents and time-varying electric fields. This is the equation that contains the displacement current.

Electromagnetic waves

Displacement current and EM waves

The displacement current is what makes electromagnetic waves possible. A time-varying $\vec{E}$ generates $\vec{B}$ (via the Ampère-Maxwell law), and a time-varying $\vec{B}$ generates $\vec{E}$ (via Faraday's law). This mutual regeneration sustains a wave that propagates through space without needing any medium or any charges.

Wave equation derivation

To derive the wave equation in free space ($\rho = 0$, $\vec{J} = 0$):

1. Take the curl of Faraday's law: $\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$
2. Substitute the Ampère-Maxwell law for $\nabla \times \vec{B}$: $\nabla \times (\nabla \times \vec{E}) = -\mu_0 \varepsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$
3. Use the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E}$, and since $\nabla \cdot \vec{E} = 0$ in free space:

$\nabla^2 \vec{E} = \mu_0 \varepsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$

An identical equation holds for $\vec{B}$. This is a wave equation with speed $c = 1/\sqrt{\mu_0 \varepsilon_0}$.

Propagation of EM waves in vacuum

In vacuum, EM waves travel at:

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^8 \text{ m/s}$

The wave is transverse: $\vec{E}$ and $\vec{B}$ are perpendicular to each other and to the propagation direction. Their magnitudes are related by $|\vec{E}| = c|\vec{B}|$.

Propagation of EM waves in media

In a linear medium with permittivity $\varepsilon$ and permeability $\mu$, the wave speed becomes:

$v = \frac{1}{\sqrt{\varepsilon \mu}}$

The index of refraction is $n = c/v = \sqrt{\varepsilon_r \mu_r}$, where $\varepsilon_r$ and $\mu_r$ are the relative permittivity and permeability. At boundaries between media, EM waves undergo reflection, refraction, and (in lossy media) absorption.

Poynting vector and energy flow

Poynting vector definition

The Poynting vector describes the directional energy flux of an electromagnetic field:

$\vec{S} = \vec{E} \times \vec{H}$

It has units of W/m² and points in the direction of energy propagation. For a plane wave in vacuum, $\vec{S}$ points along the direction of travel.

Energy density of EM fields

The total electromagnetic energy density is the sum of electric and magnetic contributions:

$u = u_E + u_B = \frac{1}{2}\varepsilon |\vec{E}|^2 + \frac{1}{2}\mu |\vec{H}|^2$

For a plane wave in vacuum, $u_E = u_B$ at every instant, so the energy is shared equally between the electric and magnetic fields.

Power flow in EM waves

The instantaneous power per unit area through a surface is $|\vec{S}|$. For sinusoidal waves, the physically meaningful quantity is usually the time-averaged Poynting vector:

$\langle \vec{S} \rangle = \frac{1}{2} \text{Re}(\vec{E} \times \vec{H}^*)$

This gives the intensity of the wave. For a plane wave in vacuum with electric field amplitude $E_0$:

$\langle S \rangle = \frac{E_0^2}{2\mu_0 c} = \frac{1}{2} c \varepsilon_0 E_0^2$

Energy conservation for EM fields is expressed by Poynting's theorem:

$-\frac{\partial u}{\partial t} = \nabla \cdot \vec{S} + \vec{J} \cdot \vec{E}$

The left side is the rate of decrease of stored energy; the right side accounts for energy flowing out (divergence of $\vec{S}$) and energy dissipated by currents ($\vec{J} \cdot \vec{E}$).

Displacement current experiments

Capacitor charging experiments

The simplest demonstration: measure the current in the wires of a charging capacitor, then use an Amperian loop that passes through the gap instead of through the wire. The magnetic field around the gap matches what you'd predict from the displacement current $\varepsilon_0 \frac{d\Phi_E}{dt}$, confirming that the displacement current produces the same magnetic effect as the conduction current in the wire.

Hertzian dipole and EM waves

Heinrich Hertz (1887) provided the first experimental confirmation of Maxwell's prediction. He used an oscillating dipole antenna to generate EM waves and a loop antenna to detect them across the room. By measuring the speed, polarization, and reflection/refraction behavior of these waves, Hertz showed they behaved exactly as Maxwell's equations (with displacement current) predicted.

Displacement current measurement techniques

Direct measurement of displacement current is challenging because the effect is small at low frequencies. Some approaches include:

• Split-ring resonators: Metamaterial structures that concentrate and enhance the displacement current at resonance, making it easier to detect.
• Modified Rogowski coils: Toroidal pickup coils placed around a region of changing electric flux to measure the magnetic field produced by the displacement current.
• Microwave cavity experiments: At GHz frequencies, displacement current effects are large enough to measure directly through their influence on cavity resonance modes.