Glossary

24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed

3 min read•june 18, 2024

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

, including light, radio waves, and X-rays, are a result of these interactions. They transport energy and momentum through space at the , playing crucial roles in communication, technology, and our understanding of the universe.

Maxwell's Equations and Electromagnetic Waves

Unification in Maxwell's equations

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  • Maxwell's equations describe the behavior of electric and magnetic fields and their interactions
    • for electric fields ∇⋅E=ρϵ0\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} relates the to the charge distribution
    • Gauss's law for magnetic fields ∇⋅B=0\nabla \cdot \mathbf{B} = 0 states that magnetic fields have no divergence and there are no magnetic monopoles
    • of induction ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} describes how a changing induces an electric field (electric generator, transformer)
    • Ampère's circuital law with Maxwell's correction ∇×B=μ0([J](https://www.fiveableKeyTerm:J)+ϵ0∂E∂t)\nabla \times \mathbf{B} = \mu_0\left(\mathbf{[J](https://www.fiveableKeyTerm:J)} + \epsilon_0\frac{\partial \mathbf{E}}{\partial t}\right) relates the magnetic field to the electric current and the changing electric field
  • The equations reveal the interconnectedness of electric and magnetic fields
    • A changing electric field generates a magnetic field (electromagnet)
    • A changing magnetic field generates an electric field (induction cooktop)
  • Maxwell's correction to introduces the term ϵ0∂E∂t\epsilon_0\frac{\partial \mathbf{E}}{\partial t} which allows for the existence of electromagnetic waves (radio waves, light)

Generation of electromagnetic waves

  • Accelerating charges generate electromagnetic waves
    • An oscillating electric charge creates a time-varying electric field (radio antenna)
    • The time-varying electric field induces a time-varying magnetic field according to Ampère's law
    • The time-varying magnetic field induces a time-varying electric field according to Faraday's law
    • This self-sustaining process results in the propagation of electromagnetic waves (radio transmission)
  • Electromagnetic waves are transverse waves where the electric and magnetic fields oscillate perpendicular to each other and to the direction of
  • Electromagnetic waves propagate through a vacuum at the speed of light [c](https://www.fiveableKeyTerm:c)[c](https://www.fiveableKeyTerm:c) and can also travel through various media (air, water, glass)
  • As electromagnetic waves propagate, they transport energy and momentum with the energy density proportional to the square of the electric and magnetic field amplitudes (solar , microwave oven)
    • The energy flow in electromagnetic waves can be described by the

Speed calculation for EM waves

  • Maxwell's equations can be used to derive the speed of electromagnetic waves in a vacuum
  • Starting with Faraday's law ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} and Ampère's law ∇×B=μ0ϵ0∂E∂t\nabla \times \mathbf{B} = \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}
  • Taking the curl of both sides of Faraday's law and substituting Ampère's law:
    1. ∇×(∇×E)=−∂∂t(∇×B)\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B})
    2. ∇×(∇×E)=−μ0ϵ0∂2E∂t2\nabla \times (\nabla \times \mathbf{E}) = -\mu_0\epsilon_0\frac{\partial^2 \mathbf{E}}{\partial t^2}
  • Using the vector identity ∇×(∇×E)=∇(∇⋅E)−∇2E\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2\mathbf{E} and Gauss's law for electric fields in a vacuum ∇⋅E=0\nabla \cdot \mathbf{E} = 0:
    • ∇2E=μ0ϵ0∂2E∂t2\nabla^2\mathbf{E} = \mu_0\epsilon_0\frac{\partial^2 \mathbf{E}}{\partial t^2} which is the for the electric field
  • The speed of the wave is given by v=1μ0ϵ0v = \frac{1}{\sqrt{\mu_0\epsilon_0}}
  • Substituting the values of the permeability of free space μ0\mu_0 and permittivity of free space ϵ0\epsilon_0, the speed of electromagnetic waves in a vacuum is c=1μ0ϵ0≈3×108 m/sc = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8 \text{ m/s} (speed of light)
    • The ratio of these electromagnetic constants defines the in free space

Electromagnetic Spectrum and Wave Properties

  • The encompasses all types of electromagnetic radiation, from radio waves to gamma rays
  • Wave propagation describes how electromagnetic waves travel through space and various media
  • Electromagnetic waves can exhibit , where the electric field oscillates in a specific direction
  • Electromagnetic radiation refers to the emission and transmission of energy in the form of electromagnetic waves

Key Terms to Review (34)

∇ · B: ∇ · B, known as the divergence of the magnetic field, is a mathematical concept that describes the density of the outward flux of a magnetic field from an infinitesimal volume around a given point. It is a fundamental quantity in the study of electromagnetism and is closely related to Maxwell's equations, which govern the behavior of electromagnetic waves.
∇ · E: ∇ · E, known as the divergence of the electric field, is a mathematical operator that describes the density of the outward flux of a vector field from an infinitesimal volume around a given point. It is a fundamental concept in the study of electromagnetism and is closely related to Maxwell's Equations, which describe the behavior of electromagnetic waves.
∇ × B: The curl of the magnetic field, denoted as ∇ × B, is a vector field that describes the circulation or rotation of the magnetic field at a given point in space. It is a fundamental concept in electromagnetism and is one of the four Maxwell's equations, which describe the relationships between electric and magnetic fields.
∇ × E: The term ∇ × E represents the curl of the electric field vector E, which quantifies the rotation or swirling motion of the electric field at a point in space. This concept is crucial in understanding how changing magnetic fields can induce electric fields, as described by one of Maxwell’s equations. Essentially, it illustrates the relationship between electric and magnetic fields, highlighting how a time-varying magnetic field can create a non-conservative electric field.
Ampère's Law: Ampère's law is a fundamental principle in electromagnetism that describes the relationship between an electric current and the magnetic field it creates. It establishes a quantitative link between the circular magnetic field generated around a current-carrying conductor and the magnitude of the electric current flowing through it.
C: The speed of light, a fundamental constant in physics that represents the maximum speed at which all matter and energy can travel through a vacuum. It is a crucial parameter in Maxwell's equations and the theory of electromagnetic waves.
Direction of polarization: Direction of polarization refers to the orientation of the electric field vector in an electromagnetic wave. It describes how the electric field oscillates as the wave propagates.
Displacement Current: Displacement current is a concept introduced by James Clerk Maxwell to complete his set of equations describing the relationship between electric and magnetic fields. It accounts for the time-varying electric field as a source of magnetic field, complementing the conduction current as a source of magnetic field.
Electric and magnetic fields: Electric and magnetic fields are two interdependent fields that propagate as waves through space. They form the basis of electromagnetic waves, where oscillations in one field induce oscillations in the other.
Electric Field: The electric field is a vector field that describes the force experienced by a stationary, positive test charge at any given point in space. It represents the strength and direction of the electric force exerted on a charged particle by other charges in the vicinity, and is a fundamental concept in the study of electromagnetism and the behavior of charged particles.
Electromagnetic spectrum: The electromagnetic spectrum is the range of all types of electromagnetic radiation, which includes visible light, radio waves, microwaves, infrared, ultraviolet, X-rays, and gamma rays. This spectrum is crucial because it encompasses the various forms of energy that travel through space at the speed of light and affects many aspects of physics, including radiation, magnetism, wave production, and energy transfer.
Electromagnetic waves: Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They travel at the speed of light and do not require a medium.
Electromagnetic Waves: Electromagnetic waves are a form of energy that propagates through space and time as oscillating electric and magnetic fields. These waves are capable of transmitting energy without the need for a physical medium, and they can travel at the speed of light.
Faraday's Law: Faraday's law describes the relationship between a changing magnetic field and the electric field it induces. It states that the magnitude of the induced electromotive force (emf) in a circuit is proportional to the rate of change of the magnetic flux through the circuit.
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 way to calculate the electric field based on the distribution of electric charges.
Heinrich Hertz: Heinrich Hertz was a German physicist who provided experimental confirmation of the existence of electromagnetic waves, as predicted by James Clerk Maxwell's electromagnetic theory. Hertz's groundbreaking work on electromagnetic waves laid the foundation for the development of modern wireless communication technologies.
Ink jet printer: An inkjet printer is a type of printer that recreates digital images by propelling droplets of ink onto paper. It utilizes electrical charges and fields to control the placement and movement of the ink.
J: J is a fundamental physical quantity that represents the amount of energy transported per unit time, or power. It is a crucial concept in the study of physics, particularly in the context of Hooke's Law, wave energy, and electromagnetic waves.
James Clerk Maxwell: James Clerk Maxwell was a Scottish physicist known for his groundbreaking contributions to the field of electromagnetism and kinetic theory. He is most famous for formulating a set of equations that describe how electric and magnetic fields interact, ultimately predicting the existence of electromagnetic waves. His work laid the foundation for modern physics, linking various phenomena across disciplines through a unified theoretical framework.
Magnetic Field: A magnetic field is a region in space where magnetic forces can be detected. It is a vector field that describes the magnetic influence of electric currents and magnetized materials on the space around them. The magnetic field is a fundamental concept in electromagnetism and is essential for understanding various phenomena in physics, including the behavior of ferromagnets, the motion of charged particles, and the production of electromagnetic waves.
Maxwell's Equations: Maxwell's equations are a set of four fundamental equations 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 various electromagnetic phenomena.
Polarization: Polarization is a fundamental property of electromagnetic waves, including light, that describes the orientation of the electric field oscillations within the wave. It is a crucial concept that underlies many important phenomena in the fields of static electricity, electromagnetism, and optics.
Poynting Vector: The Poynting vector is a mathematical quantity that describes the directional energy flux (the flow of energy per unit area) of an electromagnetic field. It represents the rate of energy transfer per unit area of an electromagnetic wave.
Radiation: Radiation refers to the emission and propagation of energy in the form of waves or particles through space or a medium. It is a fundamental concept that underpins various physical phenomena and processes, including heat transfer, electromagnetic waves, and nuclear reactions.
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 of nature that plays a crucial role in various areas of physics, from Maxwell's equations to general relativity.
Tesla: The tesla (T) is the SI unit of magnetic field strength or magnetic flux density. It measures how much force a magnetic field exerts on moving charges or current-carrying wires.
Tesla: The tesla (T) is the unit of magnetic flux density or magnetic induction in the International System of Units (SI). It is named after the Serbian-American inventor and electrical engineer Nikola Tesla, who made significant contributions to the design of the modern alternating-current (AC) electrical supply system.
Wave Equation: The wave equation is a fundamental mathematical equation that describes the propagation of waves, such as sound waves, electromagnetic waves, and water waves, through a medium. It relates the spatial and temporal variations of the wave amplitude or displacement to the properties of the medium through which the wave is traveling.
Wave Impedance: Wave impedance, also known as characteristic impedance, is a fundamental property of waves that describes the ratio of the electric field to the magnetic field, or the ratio of the voltage to the current, in a wave propagating through a medium. It is a measure of the opposition a medium presents to the flow of electromagnetic energy.
Wave Propagation: Wave propagation refers to the movement and transmission of waves through a medium or space. It describes how waves, such as electromagnetic waves or sound waves, travel and spread out from their source, transferring energy without the movement of the medium itself.
Weber: The weber (symbol: Wb) is the unit of magnetic flux in the International System of Units (SI). It is named after the German physicist Wilhelm Eduard Weber. The weber is a fundamental unit that is used to quantify the amount of magnetic flux present in a magnetic field, and it plays a crucial role in understanding various electromagnetic phenomena.
ε₀: ε₀, also known as the permittivity of free space or the electric constant, is a fundamental physical constant that represents the capability of free space or a vacuum to support an electric field. It is a crucial parameter in Maxwell's equations, which describe the behavior of electromagnetic waves and fields.
μ₀: μ₀, also known as the permeability of free space or the vacuum permeability, is a fundamental physical constant that represents the magnetic permeability of free space or a vacuum. It is a crucial parameter in Maxwell's equations, which describe the behavior of electromagnetic fields and the propagation of electromagnetic waves.
ρ (Rho): ρ, or rho, is a symbol commonly used in physics and mathematics to represent various physical quantities, including density, resistivity, and the Stefan-Boltzmann constant. This versatile symbol is particularly relevant in the contexts of fluid mechanics, wave theory, and electromagnetism.
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