Name: transverse fields in electromagnetic waves
Uploaded: 2023-04-30T12:46:03.000Z
Duration: 1 min 30 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physics

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Waves and Their Applications in Technologies for Information Transfer

PS.Phy.7.1

Plane Electromagnetic Waves I

Stellen Sie sich eine xyz-Koordinate vor, die durch eine zur yz-Ebene parallele Ebene in zwei unterschiedliche Bereiche unterteilt ist.
Auf der linken Seite liegt ein gleichmäßiges elektrisches und magnetisches Feld in +y- und +z-Richtung vor, während die rechte Seite feldfrei ist.
Diese Grenzebene, die als Wellenfront bezeichnet wird, bewegt sich mit konstanter Geschwindigkeit in +x-Richtung.
Solche Wellen, bei denen die Felder zu jedem Zeitpunkt über eine Ebene senkrecht zur Ausbreitungsrichtung gleichmäßig sind, werden als ebene Wellen bezeichnet.
Diese Wellen erfüllen die Maxwell-Gleichungen. Stellen Sie sich einen rechteckigen Gaußschen Kasten vor.
Da sich das elektrische Feld in y-Richtung befindet, sind die elektrischen Flüsse für die Oberfläche entlang der x-Richtung und z-Richtung Null.
Für die obere Oberfläche ist das Flussmittel gegenüber der unteren Oberfläche, so dass kein Nettoflussmittel entsteht.
Da die Box keine elektrische Ladung umschließt, ist sowohl der elektrische als auch der magnetische Fluss Null.
Das Gaußsche Gesetz gilt also auch für elektromagnetische Strahlung.
Daher stehen die elektrischen und magnetischen Feldvektoren senkrecht zueinander und zur Ausbreitungsrichtung, was die transversale Natur zeigt.

<ol>
	<li>Young, H.D. und Freedman, R.A. (2012). Universitätsphysik mit moderner Physik. San Francisco, Kalifornien: Pearson. Abschnitt 32.2; Seiten 1055–1056.</li>
	<li>OpenStax. (2019). Universitätsphysik Vol. 2. [Webversion]. Abgerufen von <a href="https://openstax.org/details/books/university-physics-volume-2">https://openstax.org/details/books/university-physics-volume-2</a>; Abschnitt 16.2; Seiten 706–707.</li>
</ol>

ebene der elektromagnetischen wellen i

Die Existenz von kombinierten elektrischen und magnetischen Feldern, die sich als elektromagnetische (EM) Wellen durch den Raum bewegen, ist die bedeutendste Vorhersage von Maxwells Gleichungen. Da Maxwells Gleichungen im freien Raum gelten, benötigen die vorhergesagten elektromagnetischen Wellen kein Medium für ihre Ausbreitung. Eine EM-Welle besteht aus einem elektrischen Feld, definiert als die Kraft pro Ladung auf eine ruhende Ladung, und einem magnetischen Feld, das die Kraft pro Ladung auf eine bewegte Ladung ist.

Es wird angenommen, dass das EM-Feld, nur eine Funktion der x-Koordinate und der Zeit ist. Daher wird die y-Komponente des elektrischen Feldes als Ey(x, t) bezeichnet und die z-Komponente des magnetischen Feldes als Bz(x, t). Aufgrund der Annahme des freien Raums gibt es keine freien Ladungen oder Ströme, und daher werden die in Maxwells Gleichung eingeschlossene Ladung und der Strom auf Null gesetzt.

Die Anwendbarkeit des Gaussschen Gesetzes für elektrische Felder wird durch Annahme einer rechteckigen Gauss'schen Fläche mit einem quadratischen Querschnitt der Seitenlänge l und einer dritten Seite mit Länge Δx untersucht. Die y-Komponente des elektrischen Feldes ist auf den oberen und unteren Seiten des Kastens gleich. Daher heben sich die Flüsse auf beiden Seiten gegenseitig auf. Ähnlich wird der Nettofluss aus der z-Komponente des elektrischen Feldes durch die beiden lateralen Seiten ebenfalls aufgehoben. Jeglicher Nettofluss durch die Oberfläche kommt daher ausschließlich von der x-Komponente des elektrischen Feldes. Da das elektrische Feld keine y- oder z-Abhängigkeit hat, ist Ex(x, t) über die Fläche eines Kastens mit der Fläche A konstant und hat möglicherweise einen unterschiedlichen Wert von Ex(x + Δx, t), der über die gegenüberliegende Fläche des Kastens konstant ist. Das Anwenden des Gaussschen Gesetzes ergibt den Nettofluss der x-Komponente.

<img alt="Equation1" src="/files/ftp_upload/13822/13822_Equation_PT_1.svg" style="text-align: center; height: 30px;" />

Da die eingeschlossene Ladung q_enc gleich Null ist, ist auch der Nettofluss der Komponente Null, und die obige Gleichung impliziert Ex(x, t) = Ex(x + Δx, t) für beliebiges Δx. Eine solche Komponente Ex(x, t) wäre kein Teil einer entlang der x-Achse propagierenden elektromagnetischen Welle, daher ist Ex(x, t) = 0 für diese Welle. Daher sind die einzigen von Null verschiedenen Komponenten des elektrischen Feldes Ey(x, t) und Ez(x, t), die senkrecht zur Ausbreitungsrichtung der Welle stehen.

Ein ähnliches Argument wird angewendet, indem E für B eingesetzt wird und das Gausssche Gesetz für Magnetismus anstelle des Gaussschen Gesetzes für elektrische Felder verwendet wird. Dies zeigt, dass das B-Feld auch senkrecht zur Ausbreitungsrichtung der Welle steht. Die elektromagnetische Welle ist daher eine transversale Welle, deren oszillierende elektrische und magnetische Felder senkrecht zu ihrer Ausbreitungsrichtung stehen.

Ebene der elektromagnetischen Wellen I

Die Existenz von kombinierten elektrischen und magnetischen Feldern, die sich als elektromagnetische (EM) Wellen durch den Raum ausbreiten, ist die bedeutendste Vorhersage der ...

Education

Lehre

JoVE Core

Physik

Unit 1

Einheit 1

Electromagnetic Waves

Elektromagnetische Wellen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

transverse fields in electromagnetic waves

Electromagnetic waves are made of electric and magnetic fields that move through space together. Maxwell's equations predict these waves, and they can travel in free space without a medium. The electric field acts on a stationary charge, while the magnetic field acts on a moving charge.

For a wave traveling along the x-axis, the fields are written as Ey(x, t) and Bz(x, t). In free space, there are no free charges or currents, so the charge enclosed and current terms in Maxwell's equations are set to zero. This lets us test which field components can remain in the wave.

To apply Gauss's law for electric fields, a rectangular Gaussian surface with a square cross-section of side l and a third side of length Δx is used. The y-component of the electric field is the same on the top and bottom sides, so those fluxes cancel. The z-component through the two lateral sides also cancels, so any net flux comes only from Ex.

Because Ex depends only on x and time, it is constant over one face of the box and may have a different value on the opposite face. Gauss's law gives zero net flux because the enclosed charge is zero. That means Ex(x, t) = Ex(x + Δx, t) for any Δx, so Ex(x, t) is not part of the wave and must be zero. The electric field components left in the wave are Ey(x, t) and Ez(x, t), both perpendicular to the direction of travel.

A similar argument applies to the magnetic field by using Gauss's law for magnetism instead of Gauss's law for electric fields. This shows that the magnetic field is also perpendicular to the direction of propagation. An electromagnetic wave is therefore a transverse wave, with oscillating electric and magnetic fields at right angles to the wave's motion.

Transverse Fields in Electromagnetic Waves

transverse-fields-in-electromagnetic-waves

electromagnetic waves

Maxwell’s Equations Behind EM Waves

Maxwell’s equations explain how electric and magnetic fields work together to create electromagnetic waves. James Clerk Maxwell brought together the ideas of Oersted, Coulomb, Gauss, and Faraday and built a single theory of electromagnetism. Together with the Lorentz force law, his equations describe the laws of electricity and magnetism. A key part of Maxwell’s work is the symmetry in the equations. Faraday’s law shows that a changing magnetic field produces an electric field. Maxwell added...

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Heinrich Hertz used sparks and tuned circuits to show that electromagnetic waves could be produced in the lab. Starting in 1887, the German physicist ran experiments that confirmed electromagnetic waves exist and travel at the speed of light. Hertz built an alternating-current RLC circuit. RLC stands for resistor, inductor, and capacitor. The circuit was set to a known resonant frequency and connected to a loop of wire. High voltage across the gap in the loop created sparks. Those sparks gave...

the electromagnetic spectrum

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Electromagnetic waves are grouped by wavelength and frequency. This set of waves is called the electromagnetic spectrum. It includes radio waves, microwaves, infrared, visible light, X-rays, and gamma rays. Radio waves have wavelengths from millimeters to kilometers. They are often used for audio communication, such as radios. In many cases, they come from alternating current in the wires of a broadcast antenna. Radio waves also carry AM and FM radio, cellular phone signals, and TV signals. <br...

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Maxwell's equations show how electric and magnetic fields behave in a vacuum, or empty space. They are written as differential equations and describe fields created by source charges, whether the charges are still or moving. These fields also act on a test charge, and its motion can be found with the right boundary conditions. The key question in empty space is what happens when there are no source charges. In the vacuum form of Maxwell's four equations, the first-order partial differential...

plane electromagnetic waves i

Electromagnetic waves are made of electric and magnetic fields that move through space together. Maxwell's equations predict these waves, and they can travel in free space without a medium. The electric field acts on a stationary charge, while the magnetic field acts on a moving charge. For a wave traveling along the x-axis, the fields are written as Ey(x, t) and Bz(x, t). In free space, there are no free charges or currents, so the charge enclosed and current terms in Maxwell's equations are...

plane electromagnetic waves ii

Faraday’s Law in a Plane Wave

Faraday’s law can be used to link the electric and magnetic fields in a plane electromagnetic wave. Here, the wavefront moves in the x-direction at a constant speed. The goal is to show how the field strengths must be related for the wave to stay consistent with Faraday’s law. To apply the law, a rectangle of width a is placed with its area vector in the positive z-direction. The line integral is taken counterclockwise around the rectangle. One side lies outside the wavefront, so the electric...

propagation speed of electromagnetic waves

propagation-speed-of-electromagnetic-waves

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Electromagnetic waves follow Ampere’s law when there is no conduction current. A plane wavefront moves in the positive x-direction. To analyze it, a rectangle is placed in the xz-plane with its area vector pointing in the positive y-direction. The line integral around the rectangle is taken counterclockwise. The magnetic field is zero on some sides and perpendicular on others. It is parallel to one side, and that side gives a non-zero contribution to the integral. Because the left side of...

electromagnetic waves in matter

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Electromagnetic waves can travel through matter as well as through vacuum. Light is one example. It can move through air, water, or glass. When an electromagnetic wave passes through a dielectric medium, the equations that describe it change. In Ampere's law, the permittivity of free space, ε0, is replaced by the permittivity of the dielectric, ε. The vacuum permeability, μ0, is also replaced by the permeability of the medium, μ. The wave speed c in vacuum is then replaced by a lower speed, v,...

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How EM Waves Transfer Energy

Electromagnetic waves transfer energy through their electric and magnetic fields. This energy can be easy to notice, like the warmth from the summer sun, or harder to feel, like the energy in gamma rays that can damage living cells. The fields in an electromagnetic wave can exert forces on charges in a system. They can move those charges and do work on them. Even if a wave is not absorbed right away, it still carries energy away from its source. If some of the wave energy is absorbed, the...

intensity of electromagnetic waves

Poynting Vector and Light Intensity

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radiation pressure: problem solving

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standing electromagnetic waves

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standing waves in a cavity

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key terms and concepts

See why electromagnetic waves are transverse. Learn how Maxwell’s equations show that electric and magnetic fields stay perpendicular to the direction of travel.

Transverse Fields in Electromagnetic Waves | JoVE K12

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Transverse Fields in Electromagnetic Waves

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