Name: faraday’s law in a plane wave
Uploaded: 2023-04-30T12:46:11.000Z
Duration: 1 min 29 s
Channel: JoVE

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Physical Science

PS.7

PS.7.a

Plane Electromagnetic Waves II

La relación entre los campos eléctrico y magnético se obtiene aplicando la ley de Faraday a un frente de onda plano que viaja en la dirección +x con una velocidad constante.
Considere un rectángulo de altura "a" que tiene un vector de área en la dirección +z.
Al integrarse alrededor del rectángulo en sentido contrario a las agujas del reloj, se encuentra que el campo eléctrico es cero a lo largo de la longitud "PQ", o perpendicular a los elementos, excepto por la longitud "RS". Solo contribuye a que la integral dé un valor distinto de cero.
Para satisfacer la ley de Faraday, debe haber un componente de campo magnético en la dirección +z, que puede proporcionar un flujo magnético distinto de cero a través del rectángulo y, por lo tanto, una derivada de tiempo distinta de cero del flujo magnético.
En el tiempo "dt", la onda se mueve una distancia "c dt". Mientras se mueve, barre un área igual a "ac dt".
Durante este intervalo de tiempo, el flujo magnético a través del rectángulo aumenta, dando la tasa de cambio del flujo magnético.
Este valor se sustituye en la ley de Faraday, lo que da como resultado una relación entre E y B en términos de la velocidad de propagación de la onda en el vacío.

<ol>
	<li>Young, H.D. y Freedman, R.A. (2012). Física Universitaria con Física Moderna. San Francisco, CA: Pearson. Sección 32.2; Página 1056.</li>
	<li>OpenStax. (2019). Física Universitaria Vol. 2. [Versión web]. Obtenido de <a href="https://openstax.org/details/books/university-physics-volume-2">https://openstax.org/details/books/university-physics-volume-2</a>; Sección 16.2; páginas 710-711.</li>
</ol>

ondas electromagnéticas planas ii

Considere una frente de onda plana viajando en la dirección x con una velocidad constante. Esta frente de onda puede ser utilizada para obtener la relación entre los campos eléctrico y magnético con la ayuda de la ley de Faraday. <img alt="Equation1" src="/files/ftp_upload/13823/13823_Equation_PT_1.svg" style="text-align: center; height: 35px;"> <img alt="Figure1" src="/files/ftp_upload/13823/13823_PT _Figure_1.svg" style="text-align: center; height: 175px;"> Para aplicar la ley de Faraday, considere un rectángulo de ancho a, como se muestra en la figura 1, cuyo vector de área está en dirección positiva z. Para resolver la integral del lado izquierdo en la ley de Faraday, integre en sentido contrario a las manecillas del reloj a lo largo del rectángulo. El campo eléctrico es cero a lo largo de una de las longitudes que está fuera de la frente de onda, mientras que el campo eléctrico es perpendicular a los otros dos elementos de longitud. Estos tres lados no contribuyen a la integral. Solo a lo largo de un lado, el campo eléctrico tiene un valor distinto de cero y es paralelo a la longitud. Este lado contribuye a la integral, dando un valor distinto de cero. <img alt="Equation2" src="/files/ftp_upload/13823/13823_Equation_PT_2.svg" style="text-align: center; height: 35px;"> Dado que el lado izquierdo de la ley de Faraday es distinto de cero, para satisfacer la ecuación debe haber una componente de campo magnético en dirección positiva z, que puede proporcionar un flujo magnético distinto de cero a través del rectángulo y, por lo tanto, una derivada temporal del flujo magnético distinta de cero. Para obtener este valor, considere que en el tiempo dt, la onda se desplaza una distancia c dt. Mientras se mueve, barrerá un área igual a ac dt. Durante este intervalo de tiempo, el flujo magnético a través del rectángulo aumenta, lo que da la tasa de cambio del flujo magnético. Este valor se sustituye en la ley de Faraday, dando una relación entre E y B en términos de la velocidad de propagación de la onda en el vacío. <img alt="Equation3" src="/files/ftp_upload/13823/13823_Equation_PT_3.svg" style="text-align: center; height: 14px;"> Esta expresión muestra que la onda es consistente con la ley de Faraday solo si la velocidad de la onda y las magnitudes de los vectores perpendiculares están relacionados, como en la ecuación anterior.

Ondas electromagnéticas planas II

Considere un frente de onda plano que viaja en la posición en la dirección x con una velocidad constante. Este frente de onda se puede utilizar para obtener la relación ...

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KEY TERMS AND CONCEPTS

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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 field there is zero. On the other two length elements, the electric field is perpendicular to the path, so they also give no contribution. Only one side has a non-zero electric field that is parallel to the length, so that side makes the integral non-zero.

Because the left side of Faraday’s law is non-zero, the magnetic side must also be non-zero. That means there must be a magnetic field component in the positive z-direction. This component gives magnetic flux through the rectangle, and its change in time must match the electric field term.

To find the flux change, consider the time interval dt. In that time, the wavefront moves a distance c dt, where c is the wave speed in vacuum. As it moves, it sweeps out an area equal to ac dt. That growing area increases the magnetic flux through the rectangle and gives the rate of change of flux.

Substituting this result into Faraday’s law gives a relationship between E and B in terms of the wave speed in vacuum. The expression shows that the plane wave satisfies Faraday’s law only when the wave speed and the magnitudes of the perpendicular electric and magnetic fields are related in this way.

Faraday’s Law in a Plane Wave

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Use Faraday’s law to relate electric and magnetic fields in a plane wave, and see how a moving wavefront sets the link with wave speed in vacuum.

Faraday’s Law in a Plane Wave | JoVE K12

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PS.7.b

Faraday’s Law in a Plane Wave

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