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16.1 Maxwell’s Equations and Electromagnetic Waves

3 min read•june 24, 2024

are the foundation of electromagnetism, unifying electric and magnetic phenomena. These four equations describe how electric and magnetic fields interact and propagate through space, forming the basis for understanding .

Electromagnetic waves are self-sustaining oscillations of electric and magnetic fields. These waves can travel through space at the , carrying energy and information. Understanding their properties is crucial for applications in communication, medicine, and technology.

Maxwell's Equations

Significance of displacement current

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  • relates magnetic fields to only considering conduction current from moving charges
  • Maxwell added the term (∂E⃗∂t\frac{\partial \vec{E}}{\partial t}) to Ampère's law due to changing electric fields
    • Made Ampère's law consistent with the
  • Displacement current term is crucial for explaining electromagnetic waves providing the link between changing electric and magnetic fields
    • Without it, electromagnetic waves would not be possible (radio waves, light, X-rays)

Application of Maxwell's equations

  • for electric fields (∮E⃗⋅dA⃗=Qencϵ0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}) relates to enclosed charge
    • Determines electric field from charge distribution (point charges, charged plates)
  • Gauss's law for magnetic fields (∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0) states that do not exist
    • lines always form closed loops
  • (∮E⃗⋅dl⃗=−dΦBdt\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}) relates induced electric fields to changing magnetic flux
    • Explains operation of transformers and generators
  • (∮B⃗⋅dl⃗=μ0Ienc+μ0ϵ0dΦEdt\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}) relates magnetic fields to electric currents and changing electric flux
    • Determines magnetic field from current distribution (wires, solenoids)

James Clerk Maxwell's contributions

  • Unified electricity and magnetism into a single theory of electromagnetism
  • Formulated the set of equations now known as Maxwell's equations
  • Predicted the existence of electromagnetic waves
  • Calculated the speed of electromagnetic waves, which matched the known

Electromagnetic Waves

Electric and magnetic field relationships

  • Electromagnetic waves are of electric and magnetic fields
  • Changing electric field creates a changing magnetic field governed by Ampère-Maxwell law
  • Changing magnetic field creates a changing electric field governed by Faraday's law
    • Mutual generation of fields allows electromagnetic waves to propagate through space (vacuum, air, materials)
  • Electromagnetic wave properties:
    • Electric and magnetic fields are perpendicular to each other and the direction of propagation
    • Speed of propagation is the speed of light (c=1μ0ϵ0c = \frac{1}{\sqrt{\mu_0 \epsilon_0}})
    • Ratio of electric to magnetic field amplitudes is EB=c\frac{E}{B} = c
  • describes the propagation of electromagnetic waves in space and time

Hertz's confirmation of electromagnetic waves

  • conducted experiments to validate Maxwell's predictions
  • Hertz used an to generate electromagnetic waves
    1. Circuit consisted of a capacitor and an inductor
    2. Rapid switching of the circuit produced high- oscillations
  • Hertz detected the electromagnetic waves using a simple receiver
    • Receiver was a loop of wire with a small gap
    • Sparks across the gap indicated the presence of electromagnetic waves
  • Hertz measured the and frequency of the waves showing the product equaled the speed of light
    • Provided strong evidence for Maxwell's electromagnetic wave theory

Electromagnetic wave characteristics

  • Poynting vector represents the energy flux density of an electromagnetic wave
  • Polarization describes the orientation of the electric field oscillations in an electromagnetic wave
  • encompasses all types of electromagnetic radiation, from radio waves to gamma rays

Key Terms to Review (26)

Ampère-Maxwell law: The Ampère-Maxwell law is an extension of Ampère's Law that incorporates the concept of displacement current, linking electric fields and magnetic fields in electromagnetic theory. It states that the magnetic field around a closed loop is proportional to the electric current passing through the loop and the rate of change of the electric field in the region, forming a crucial part of the complete set of equations that describe electromagnetism.
Ampère's Law: Ampère's law is a fundamental principle in electromagnetism that relates the magnetic field generated by an electric current to the magnitude and direction of that current. It is one of the four Maxwell's equations, which together describe the relationships between electric and magnetic fields and electric charges and currents.
Continuity Equation: The continuity equation is a fundamental principle in physics that describes the conservation of a quantity, such as mass or electric charge, within a given system. It establishes a relationship between the rate of change of a quantity and the flux of that quantity through the boundaries of the system.
Displacement Current: Displacement current is a concept in electromagnetism that describes the flow of electric charge through a changing electric field, even in the absence of a conductive path. It is a crucial component of James Clerk Maxwell's equations, which unified the theories of electricity, magnetism, and light.
Electric Currents: Electric current is the flow of electric charge through a conductive material, such as a wire or a semiconductor. It is a fundamental concept in the study of electromagnetism and is essential for understanding the behavior of electrical circuits and the propagation of electromagnetic waves.
Electric Flux: Electric flux is a measure of the total electric field passing through a given surface. It represents the number of electric field lines passing perpendicularly through a surface, and is a key concept in understanding the behavior of electric fields and charges.
Electromagnetic Spectrum: The electromagnetic spectrum is the entire range of electromagnetic radiation, from the longest wavelengths of radio waves to the shortest wavelengths of gamma rays. This spectrum encompasses all the different forms of light, including visible light, that make up the world around us and enable various scientific and technological applications.
Electromagnetic Waves: Electromagnetic waves are a type of energy that travels through space or a medium in the form of oscillating electric and magnetic fields. These waves are responsible for various phenomena, including heat transfer, magnetism, and the propagation of electric fields, and are fundamental to our understanding of Maxwell's equations and the electromagnetic spectrum.
Faraday's law: Faraday's law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This principle is crucial for understanding how magnetic fields interact with electric circuits and lays the foundation for many applications in electromagnetism.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences or cycles of a periodic phenomenon per unit of time. It is a crucial parameter in understanding various physical processes, including alternating current (AC) sources, power in AC circuits, electromagnetic waves, and the electromagnetic spectrum.
Gauss's Law: Gauss's law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total electric charge enclosed within that surface. It provides a powerful tool for calculating the electric field produced by various charge distributions.
Heinrich Hertz: Heinrich Hertz was a German physicist who made significant contributions to the field of electromagnetism, particularly known for his experimental verification of electromagnetic waves. His work laid the foundation for the development of radio technology and demonstrated that electromagnetic waves could be generated and detected, validating James Clerk Maxwell's theories. Hertz's discoveries were pivotal in understanding the nature of electromagnetic radiation and its propagation through space.
Hertz's confirmation: Hertz's confirmation refers to the experimental validation of electromagnetic waves, demonstrated by Heinrich Hertz in the late 19th century. This groundbreaking work confirmed the existence of electromagnetic radiation predicted by Maxwell's equations, showing that electric and magnetic fields could propagate through space as waves, further establishing the foundation for modern physics and telecommunications.
James Clerk Maxwell: James Clerk Maxwell was a renowned Scottish physicist who made significant contributions to the understanding of electromagnetism, the nature of light, and the foundations of modern physics. His work laid the groundwork for many of the key concepts and theories that are central to the topics of 2.4 Distribution of Molecular Speeds, 6.4 Conductors in Electrostatic Equilibrium, 11.1 Magnetism and Its Historical Discoveries, 12.3 Magnetic Force between Two Parallel Currents, 13.4 Induced Electric Fields, 13.5 Eddy Currents, and 16.1 Maxwell's Equations and Electromagnetic Waves.
Magnetic Field: A magnetic field is a region of space where magnetic forces can be detected. It is a fundamental concept in electromagnetism, describing the invisible lines of force that surround and permeate magnetic materials, electric currents, and changing electric fields. The magnetic field plays a crucial role in various topics within the study of college physics.
Magnetic Monopoles: Magnetic monopoles are hypothetical particles that possess only a single magnetic pole, either a north or a south pole, unlike the ubiquitous dipole magnets we encounter in everyday life which have both north and south poles. The concept of magnetic monopoles arises from the fundamental laws of electromagnetism and has been the subject of extensive theoretical and experimental investigations in physics.
Maxwell’s equations: Maxwell's equations are a set of four fundamental laws that describe how electric and magnetic fields interact and propagate. They form the foundation of classical electromagnetism, optics, and electric circuits.
Maxwell's Equations: Maxwell's equations are a set of four fundamental equations in electromagnetism that describe the relationships between electric and magnetic fields and electric charges and currents. These equations form the foundation of classical electromagnetism and are essential for understanding a wide range of electromagnetic phenomena.
Oscillating electric circuit: An oscillating electric circuit is a type of electrical circuit that generates alternating current (AC) by storing and releasing energy in its inductors and capacitors, leading to periodic oscillations in voltage and current. This behavior is essential in understanding how electromagnetic waves propagate, as it creates oscillating electric and magnetic fields that can travel through space.
Permittivity of free space: Permittivity of free space, denoted as $\varepsilon_0$, is a fundamental physical constant that characterizes the ability of the vacuum to permit electric field lines. It has a value of approximately $8.854 \times 10^{-12}$ F/m (farads per meter).
Self-propagating oscillations: Self-propagating oscillations are wave-like disturbances that can sustain themselves and travel through a medium without a continuous external force. This phenomenon is fundamental in understanding how electromagnetic waves, including light, can move through space as they are generated by oscillating electric and magnetic fields that influence each other.
Speed of light: The speed of light in a vacuum is the constant speed at which all electromagnetic waves propagate, approximately $3 \times 10^8$ meters per second. It is a fundamental constant in physics denoted by the symbol $c$.
Speed of Light: The speed of light is the maximum velocity at which all electromagnetic radiation, including visible light, can travel through a vacuum. It is a fundamental constant in physics that has profound implications across various topics in electromagnetism and relativity.
Wave Equation: The wave equation is a second-order linear partial differential equation that describes the propagation of waves, such as sound waves, light waves, and electromagnetic waves, through a medium or in a given space. It is a fundamental equation in the study of wave phenomena and is derived from the underlying physical principles governing the behavior of waves.
Wavelength: Wavelength is the distance between two consecutive peaks or troughs in a wave. It is a fundamental property of all types of waves, including electromagnetic waves, sound waves, and water waves. Wavelength is a crucial concept in understanding the behavior and characteristics of various wave phenomena.
Young: Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They are described by Maxwell's equations and can travel through a vacuum.
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16.2 Plane Electromagnetic Waves