1.5 Displacement current

Displacement current is a crucial concept in electromagnetism that explains the apparent flow of current in capacitors and empty space. It generalizes electric current to include time-varying electric fields, maintaining continuity where traditional conduction current fails.

Maxwell's introduction of displacement current resolved inconsistencies in Ampère's law and led to the prediction of electromagnetic waves. This concept is fundamental to understanding capacitor behavior, wave propagation, and the interplay between electric and magnetic fields in various applications.

Displacement current concept

• Displacement current is a crucial concept in electromagnetism that explains the apparent flow of current in a capacitor or in empty space
• It is a generalization of electric current to include time-varying electric fields, in addition to the motion of charges
• Displacement current maintains the continuity of current in situations where the traditional conduction current fails to do so

Capacitor charging and current

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• When a capacitor is charging, there is a current flowing in the circuit, even though no actual charges are crossing the gap between the plates
• The charging current is due to the changing electric field between the capacitor plates, which induces a displacement current
• Displacement current in a capacitor is proportional to the rate of change of the electric field between the plates

Electric field and displacement current

• Displacement current arises from a time-varying electric field, as described by Maxwell's equations
• It is defined as the rate of change of the electric displacement field $\vec{D}$, given by $\vec{J_D} = \frac{\partial \vec{D}}{\partial t}$
• The electric displacement field is related to the electric field $\vec{E}$ by the permittivity of the medium $\varepsilon$, as $\vec{D} = \varepsilon \vec{E}$

Displacement current vs conduction current

• Conduction current is the flow of electric charges in a conductor, caused by an electric field
• Displacement current is the apparent flow of current due to a changing electric field, even in the absence of moving charges
• In a closed loop, the total current (conduction + displacement) is always continuous, ensuring charge conservation

Displacement current equation

• The displacement current equation is a fundamental relation in electromagnetism that quantifies the contribution of time-varying electric fields to the total current
• It is an essential part of Maxwell's equations, which form the foundation of classical electromagnetism
• Understanding the displacement current equation is crucial for analyzing electromagnetic phenomena in various contexts

Displacement current formula

• The displacement current density $\vec{J_D}$ is given by the formula: $\vec{J_D} = \frac{\partial \vec{D}}{\partial t}$
• Here, $\vec{D}$ is the electric displacement field, which is related to the electric field $\vec{E}$ by the permittivity $\varepsilon$ of the medium
• The displacement current $I_D$ through a surface $S$ is the integral of the displacement current density over that surface: $I_D = \int_S \vec{J_D} \cdot d\vec{S}$

Permittivity and electric flux

• Permittivity $\varepsilon$ is a physical quantity that describes the ability of a medium to store electrical energy in an electric field
• It relates the electric field $\vec{E}$ to the electric displacement field $\vec{D}$ as $\vec{D} = \varepsilon \vec{E}$
• The electric flux $\Phi_E$ through a surface is the integral of the electric displacement field over that surface: $\Phi_E = \int_S \vec{D} \cdot d\vec{S}$

Displacement current derivation

• The displacement current can be derived from Maxwell's equations, specifically from Ampère's circuital law and Gauss's law for electric fields
• By taking the divergence of Ampère's law and using the continuity equation, one can show that a time-varying electric field contributes to the total current
• This contribution is the displacement current, which is necessary to maintain the conservation of charge and the consistency of Maxwell's equations

Displacement current applications

• Displacement current has numerous applications in various fields of physics and engineering, ranging from capacitor circuits to electromagnetic wave propagation
• It plays a crucial role in understanding the behavior of time-varying electromagnetic fields and their interactions with matter
• Analyzing displacement current is essential for designing and optimizing electronic devices, antennas, and communication systems

Displacement current in capacitors

• In a charging or discharging capacitor, the displacement current between the plates is equal to the conduction current in the connecting wires
• This ensures the continuity of current in the circuit, even though no charges are physically moving across the capacitor gap
• The displacement current in a capacitor is proportional to the rate of change of the voltage across the plates

Displacement current in vacuum

• In a vacuum, where there are no free charges, displacement current can still exist due to time-varying electric fields
• This is a crucial insight from Maxwell's equations, which predicts the existence of electromagnetic waves in vacuum
• The displacement current in vacuum is proportional to the rate of change of the electric field

Displacement current in dielectrics

• In dielectric materials, the displacement current is modified by the presence of bound charges, which polarize in response to an applied electric field
• The displacement current in a dielectric is related to the rate of change of the electric displacement field $\vec{D}$, which includes the effect of polarization
• Understanding displacement current in dielectrics is important for analyzing capacitors with dielectric materials and the propagation of electromagnetic waves in media

Ampère's circuital law

• Ampère's circuital law is a fundamental relation in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop
• It is one of Maxwell's equations and plays a crucial role in understanding the generation and properties of magnetic fields
• The original form of Ampère's law, without the displacement current term, is inconsistent with the continuity equation and requires modification

Ampère's law without displacement current

• The original form of Ampère's circuital law states that the integral of the magnetic field $\vec{B}$ around a closed loop is equal to $\mu_0$ times the total current $I$ enclosed by the loop
• Mathematically, $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$, where $\mu_0$ is the permeability of free space
• This form of Ampère's law does not account for the possibility of a time-varying electric field generating a magnetic field

Inconsistency with continuity equation

• The continuity equation, which expresses the conservation of electric charge, requires that the divergence of the current density $\vec{J}$ is equal to the negative rate of change of the charge density $\rho$
• Mathematically, $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$
• Ampère's law without the displacement current term violates the continuity equation in situations where there is a time-varying electric field, such as in a charging capacitor

Generalized Ampère's law with displacement current

• To resolve the inconsistency and make Ampère's law compatible with the continuity equation, Maxwell introduced the concept of displacement current
• The generalized Ampère's law, also known as the Ampère-Maxwell law, includes the displacement current term $\varepsilon_0 \frac{\partial \vec{E}}{\partial t}$
• The complete equation becomes $\oint \vec{B} \cdot d\vec{l} = \mu_0 (I + \varepsilon_0 \frac{d\Phi_E}{dt})$, where $\Phi_E$ is the electric flux through the loop

Maxwell's equations

• Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions with matter
• They are the foundation of classical electromagnetism and provide a unified framework for understanding a wide range of electromagnetic phenomena
• The four equations are Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of induction, and the Ampère-Maxwell law

Gauss's law for electric fields

• Gauss's law for electric fields states that the total electric flux through any closed surface is equal to the total electric charge enclosed by the surface divided by the permittivity of free space $\varepsilon_0$
• Mathematically, $\oint \vec{E} \cdot d\vec{S} = \frac{Q}{\varepsilon_0}$, where $Q$ is the total electric charge enclosed
• This law relates the electric field to the distribution of electric charges

Gauss's law for magnetic fields

• Gauss's law for magnetic fields states that the total magnetic flux through any closed surface is always zero
• Mathematically, $\oint \vec{B} \cdot d\vec{S} = 0$
• This law implies that magnetic monopoles do not exist, and magnetic field lines always form closed loops

Faraday's law of induction

• Faraday's law of induction states that a time-varying magnetic flux through a loop induces an electromotive force (EMF) in the loop, which is equal to the negative rate of change of the magnetic flux
• Mathematically, $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux
• This law describes the generation of electric fields by changing magnetic fields

Ampère-Maxwell law

• The Ampère-Maxwell law is a generalization of Ampère's circuital law that includes the displacement current term
• It states that the magnetic field around a closed loop is related to the total current (conduction current + displacement current) passing through the loop
• Mathematically, $\oint \vec{B} \cdot d\vec{l} = \mu_0 (I + \varepsilon_0 \frac{d\Phi_E}{dt})$, where $I$ is the conduction current and $\Phi_E$ is the electric flux
• This law describes the generation of magnetic fields by electric currents and time-varying electric fields

Electromagnetic waves

• Electromagnetic (EM) waves are oscillating disturbances in the electric and magnetic fields that propagate through space at the speed of light
• They are a fundamental consequence of Maxwell's equations and play a crucial role in numerous applications, from radio communication to optical imaging
• Understanding the properties and behavior of EM waves is essential for many areas of physics and engineering

Displacement current and EM waves

• The displacement current, introduced by Maxwell, is a key concept in the theory of electromagnetic waves
• It allows for the generation of magnetic fields by time-varying electric fields, even in the absence of conduction currents
• The interplay between the electric and magnetic fields, coupled through the displacement current, gives rise to self-sustaining EM waves that propagate through space

Wave equation derivation

• The wave equation for electromagnetic waves can be derived from Maxwell's equations by taking the curl of Faraday's law and the Ampère-Maxwell law
• This leads to second-order partial differential equations for the electric and magnetic fields, which have the form of wave equations
• The wave equations describe the spatial and temporal evolution of the electric and magnetic fields in an EM wave

Propagation of EM waves in vacuum

• In a vacuum, electromagnetic waves propagate at the speed of light, denoted by $c$, which is approximately $3 \times 10^8$ m/s
• The speed of light in vacuum is related to the permittivity $\varepsilon_0$ and permeability $\mu_0$ of free space by $c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$
• The electric and magnetic fields in an EM wave in vacuum are perpendicular to each other and to the direction of propagation, forming a transverse wave

Propagation of EM waves in media

• When electromagnetic waves propagate through a medium, their speed and other properties are affected by the medium's permittivity $\varepsilon$ and permeability $\mu$
• The speed of an EM wave in a medium is given by $v = \frac{1}{\sqrt{\varepsilon \mu}}$, which is generally lower than the speed of light in vacuum
• The presence of the medium can also cause the EM wave to be partially reflected, refracted, or absorbed, depending on the material properties

Poynting vector and energy flow

• The Poynting vector is a quantity that represents the direction and magnitude of energy flow in an electromagnetic field
• It is a crucial concept for understanding the propagation and distribution of energy in EM waves and is widely used in antenna theory, optical physics, and other areas
• The Poynting vector is closely related to the energy density of the electromagnetic field and the power transmitted by EM waves

Poynting vector definition

• The Poynting vector $\vec{S}$ is defined as the cross product of the electric field $\vec{E}$ and the magnetic field $\vec{H}$, i.e., $\vec{S} = \vec{E} \times \vec{H}$
• It has units of power per unit area (W/m²) and points in the direction of energy flow
• The magnitude of the Poynting vector represents the intensity of the electromagnetic energy flow at a given point

Energy density of EM fields

• The energy density of an electromagnetic field is the sum of the electric and magnetic field energy densities
• The electric field energy density is given by $u_E = \frac{1}{2} \varepsilon |\vec{E}|^2$, where $\varepsilon$ is the permittivity of the medium
• The magnetic field energy density is given by $u_B = \frac{1}{2} \mu |\vec{H}|^2$, where $\mu$ is the permeability of the medium
• The total electromagnetic energy density is $u = u_E + u_B$

Power flow in EM waves

• The power flow in an electromagnetic wave is described by the Poynting vector
• The instantaneous power per unit area flowing through a surface perpendicular to the Poynting vector is given by the magnitude of $\vec{S}$
• The average power flow, or intensity, of an EM wave is given by the time average of the Poynting vector, $\langle \vec{S} \rangle = \frac{1}{2} \text{Re}(\vec{E} \times \vec{H}^*)$, where $\vec{H}^*$ is the complex conjugate of $\vec{H}$

Displacement current experiments

• Experimental verification of the displacement current concept has been crucial for validating Maxwell's equations and the theory of electromagnetic waves
• Various experiments have been designed and conducted to measure and demonstrate the existence of displacement current in different contexts
• These experiments have not only confirmed the theoretical predictions but also led to practical applications in electronics, telecommunications, and other fields

Capacitor charging experiments

• One of the most straightforward ways to observe displacement current is through capacitor charging experiments
• By measuring the current in the connecting wires of a charging capacitor, one can indirectly detect the presence of displacement current between the capacitor plates
• The displacement current in the capacitor is equal to the conduction current in the wires, ensuring the continuity of current in the circuit

Hertzian dipole and EM waves

• The Hertzian dipole, named after Heinrich Hertz, is a simple antenna consisting of a center-fed dipole that can generate and detect electromagnetic waves
• Hertz used this setup to experimentally demonstrate the existence of EM waves, as predicted by Maxwell's equations
• By measuring the electromagnetic fields at a distance from the dipole, Hertz confirmed the presence of displacement current and the propagation of EM waves in free space

Displacement current measurement techniques

• Various techniques have been developed to directly measure displacement current in different scenarios
• One approach is to use a split-ring resonator, which is a metamaterial structure that can enhance the displacement current and make it easier to detect
• Another technique involves using a modified Rogowski coil, which is a toroidal coil that can measure the magnetic field induced by the displacement current
• These and other measurement methods have provided valuable insights into the behavior of displacement current and its role in electromagnetic phenomena