Displacement Current - Source Of Displacement Current & Modified Ampere Law

Displacement Current - Source of Displacement Current & Modified Ampere Law

Introduction

In the previous lecture, we discussed Ampere’s law and its limitations.
Today, we will learn about the concept of displacement current and how it acts as a source of magnetic fields.
We will also study the modified Ampere’s law, which includes the effects of displacement current.

What is Displacement Current?

Displacement current is a concept introduced by Maxwell to address the limitations of Ampere’s law.
According to Maxwell, when electric charges in a capacitor start to flow, a changing electric field is produced between the capacitor plates.
This changing electric field induces a magnetic field that behaves like a current.
This “current” is called displacement current.

Source of Displacement Current

Displacement current arises from the changing electric field between capacitor plates.
It does not involve the motion of charges, unlike conduction current.
It is purely a consequence of changing electric fields and their associated magnetic fields.
Displacement current helps to explain various electromagnetic phenomena, such as electromagnetic wave propagation.

Modified Ampere’s Law

Ampere’s law, in its original form, does not account for the effects of displacement current.
The modified Ampere’s law includes the effects of both conduction current and displacement current.
It states that the line integral of the magnetic field around a closed loop is equal to the sum of the conduction current and displacement current passing through the loop.
Mathematically, the modified Ampere’s law is given by: ∮ B · dl = μ₀(I_conduction + I_displacement)

Relation between Displacement Current and Electric Field

Displacement current is directly related to the rate of change of electric field with respect to time.
Mathematically, the relation between displacement current and electric field is given by: I_displacement = ε₀(dϕE/dt) where I_displacement is the displacement current, ε₀ is the permittivity of free space, and (dϕE/dt) is the rate of change of electric flux.

Example: Displacement Current Calculation

Consider a capacitor connected to an AC power supply with an alternating voltage of V = V₀*sin(ωt).
The plate separation of the capacitor is d, and the area of the plates is A.
Find the displacement current flowing through the capacitor when the voltage is V₀*sin(ωt).
Solution:
- The electric field between the plates is given by E = V/d.
- The rate of change of electric field with respect to time is (dE/dt) = (dV/dt)/d.
- The displacement current is given by: I_displacement = ε₀(dϕE/dt) = ε₀(dAdE/dt) = ε₀AdV/dt

Relationship between Displacement Current and Conductivity

Displacement current and conduction current are two distinct components of the total current in a circuit.
Displacement current is associated with the changing electric fields, while conduction current is due to the flow of charges through a conducting medium.
In insulators, where conduction current is negligible, displacement current is the dominant component.
In conductors, where conduction current is significant, displacement current may be relatively small.

Implications of Displacement Current

The concept of displacement current extends Ampere’s law to incorporate the effects of changing electric fields.
It explains the behavior of electromagnetic waves and how they can propagate through space.
Displacement current provides a deeper understanding of electromagnetism and helps in the development of various technologies such as wireless communication and radar systems.

Summary

Displacement current arises from the changing electric field between the plates of a capacitor.
It acts as a source of magnetic fields and is included in the modified Ampere’s law.
Displacement current is directly related to the rate of change of electric field with respect to time.
It coexists with conduction current and is dominant in insulating materials.
Displacement current plays a crucial role in the propagation of electromagnetic waves and various technological applications.

Displacement Current and Maxwell’s Equations

Maxwell’s equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields.
Displacement current plays a crucial role in two of these equations: Gauss’s law for electric fields and Faraday’s law of electromagnetic induction.
These equations connect the behavior of electric and magnetic fields and form the foundation of classical electromagnetism.

Gauss’s Law for Electric Fields - Including Displacement Current Term

Gauss’s law states that the electric flux through a closed surface is proportional to the total electric charge enclosed within the surface.
With the inclusion of the displacement current term, the modified Gauss’s law is given by: ∮ E · dA = (1/ε₀)∫ ρdV + (1/ε₀)(d/dt)∫ BdA
This equation shows that the electric flux is not only related to the charge density (ρ) but also to the rate of change of magnetic flux through the surface.

Faraday’s Law of Electromagnetic Induction - Including Displacement Current Term

Faraday’s law states that a changing magnetic field induces an electromotive force (EMF) in a conductor.
With the inclusion of the displacement current term, the modified Faraday’s law is given by: ∮ E · dl = -d/dt ∫ BdA
This equation shows that the induced EMF is not only due to the changing magnetic field but also the changing electric flux through the loop.

Example: Induced EMF in a Solenoid

Consider a solenoid with N turns, area A, and a changing magnetic field B(t).
According to Faraday’s law, the induced EMF in the solenoid is given by: ε = -N(dB/dt)A
This equation shows that the induced EMF is proportional to the rate of change of magnetic field and the number of turns in the solenoid.

Ampere-Maxwell Law - Including Displacement Current Term

The Ampere-Maxwell law relates the circulating magnetic field around a closed loop to the conduction current and displacement current passing through the loop.
With the inclusion of the displacement current term, the modified Ampere-Maxwell law is given by: ∮ B · dl = μ₀(I_conduction + I_displacement) + μ₀ε₀(d/dt)∫ EdA
This equation includes the effects of changing electric fields and provides a more complete description of the magnetic field generation.

Example: Calculation of Magnetic Field using Ampere-Maxwell Law

Consider a long straight wire carrying a current I and a parallel plate capacitor with a voltage V(t) applied across its plates.
The Ampere-Maxwell law can be used to find the magnetic field between the plates of the capacitor due to the conduction and displacement currents.

Resistance and Reactance

Resistance (R) is a measure of the opposition to the flow of electric current in a circuit.
Reactance (X) is a measure of the opposition to the flow of alternating current due to capacitance or inductance.
In a purely resistive circuit, the reactance is zero, and the current and voltage are in phase.
In a circuit with capacitance, the reactance is negative (capacitive reactance), and the current leads the voltage.
In a circuit with inductance, the reactance is positive (inductive reactance), and the current lags behind the voltage.

Impedance and Phase Angle

Impedance (Z) is the total opposition to the flow of alternating current in a circuit, including both resistance and reactance.
It is a complex quantity and is given by Z = R + jX, where j is the imaginary unit.
The phase angle (θ) is the phase difference between the current and voltage in an AC circuit.
It is related to the reactance and resistance by the equation tan(θ) = X/R.

Example: Calculating Impedance and Phase Angle

Consider a circuit with a resistor of resistance R and a capacitor of capacitance C connected in series.
The impedance of the circuit is given by Z = √(R² + (1/(ωC))²), where ω is the angular frequency of the AC current.
The phase angle is given by θ = tan^(-1)(1/(ωRC)).
Using these equations, we can calculate the impedance and phase angle for different values of R, C, and ω.

Applications of Displacement Current

Displacement current finds applications in various fields, including:
- Wireless communication: Displacement current plays a crucial role in the operation of antennas and propagation of signals.
- Radar systems: Displacement current helps in the detection and tracking of objects using electromagnetic waves.
- Electric circuits: Understanding displacement current is necessary for the analysis and design of complex circuits involving capacitors and changing electric fields.

Applications of Displacement Current (Continued)

Electromagnetic wave propagation: Displacement current is essential for the generation and propagation of electromagnetic waves, such as radio waves, microwaves, and light.
Electric field measurement: Displacement current allows for accurate measurement of electric fields in various applications, including high voltage systems and medical devices.
Capacitive sensors: Displacement current is utilized in capacitive sensors, which detect changes in capacitance and are used in touchscreens, proximity detectors, and humidity sensors.
Electromagnetic compatibility: Understanding displacement current helps engineers design electronic systems to minimize electromagnetic interference and ensure compatibility between devices.
Advanced technologies: Displacement current plays a role in the development of advanced technologies such as computer chips, wireless power transfer, and particle accelerators.

Maxwell’s Equations - Summarized

Maxwell’s equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields.
In integral form, the summarized Maxwell’s equations are:
1. Gauss’s law for electric fields: ∮ E · dA = (1/ε₀)∫ ρdV + (1/ε₀)(d/dt)∫ BdA
2. Gauss’s law for magnetic fields: ∮ B · dA = 0
3. Faraday’s law of electromagnetic induction: ∮ E · dl = -d/dt ∫ BdA
4. Ampere-Maxwell law: ∮ B · dl = μ₀(I_conduction + I_displacement) + μ₀ε₀(d/dt)∫ EdA

Wave Equation and Speed of Electromagnetic Waves

From Maxwell’s equations, we can derive the wave equation, which describes the behavior of electromagnetic waves.
The wave equation for electromagnetic waves in vacuum is: ∇²E - (1/c²)(∂²E/∂t²) = 0, where c is the speed of light in vacuum.
The speed of electromagnetic waves is given by: c = 1/√(ε₀μ₀), where ε₀ is the permittivity of free space and μ₀ is the permeability of free space.
The speed of electromagnetic waves in vacuum is approximately 3 × 10^8 m/s.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all possible frequencies of electromagnetic waves.
It encompasses various types of waves, including radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
Each type of wave has a specific range of frequencies and wavelengths, which have different applications and interaction characteristics.

Electromagnetic Radiation and Photons

Electromagnetic waves can be viewed as streams of particles called photons.
Each photon carries energy and momentum, and the interaction of photons with matter results in various phenomena, such as absorption, transmission, and reflection.
The energy of a photon is given by E = hf, where h is Planck’s constant and f is the frequency of the electromagnetic wave.

Electromagnetic Waves and Polarization

Electromagnetic waves can exhibit different types of polarization, which refers to the orientation of the electric field vector.
Linear polarization: The electric field vector oscillates in a single plane.
Circular polarization: The electric field vector rotates in a circular pattern.
Elliptical polarization: The electric field vector traces out an elliptical pattern.
The polarization of electromagnetic waves has important implications in various applications, including polarized sunglasses and communication systems.

Electromagnetic Waves and Diffraction

Diffraction is the bending and spreading of waves when they encounter obstacles or pass through narrow openings.
Electromagnetic waves undergo diffraction phenomena, which can be observed in various real-world scenarios, such as the diffraction of light around a door frame or the diffraction patterns produced by a CD.
Diffraction allows for the resolution of objects and the formation of interference patterns.

Electromagnetic Radiation and its Health Effects

Electromagnetic radiation, particularly in the form of ionizing radiation (X-rays and gamma rays), can have harmful effects on biological organisms.
Proper safety measures and regulations are in place to protect individuals from excessive exposure to ionizing radiation.
Non-ionizing radiation, such as radio waves and visible light, has less harmful effects and is widely used in communication and other applications.

Conclusion

Displacement current is a concept introduced by Maxwell to address the limitations of Ampere’s law.
It arises from changing electric fields and acts as a source of magnetic fields.
The modified Ampere’s law incorporates displacement current and provides a more complete understanding of electromagnetic phenomena.
Displacement current plays a crucial role in the generation and propagation of electromagnetic waves.
Understanding displacement current is essential for various technological applications and the development of advanced systems.

Summary

Displacement current arises from changing electric fields and acts as a source of magnetic fields.
It is included in the modified Ampere’s law, which provides a more complete description of electromagnetic phenomena.
Displacement current explains the behavior of electromagnetic waves and their propagation through space.
It has applications in various fields, including wireless communication, radar systems, and electric circuits.
Maxwell’s equations and the wave equation form the foundation of classical electromagnetism and describe the behavior of electric and magnetic fields.