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

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Plane Electromagnetic Waves II

La relation entre les champs électriques et magnétiques est obtenue en appliquant la loi de Faraday à un front d'onde plan se déplaçant dans la direction +x avec une vitesse constante.
Considérons un rectangle de hauteur « a » ayant un vecteur aire dans la direction +z.
Lors de l’intégration autour du rectangle dans le sens inverse des aiguilles d’une montre, le champ électrique est égal à zéro sur la longueur « PQ », ou perpendiculaire aux éléments, à l’exception de la longueur « RS ». Il ne contribue qu’à l’intégrale en donnant une valeur non nulle.
Pour satisfaire la loi de Faraday, il doit y avoir une composante de champ magnétique dans la direction +z, qui peut fournir un flux magnétique non nul à travers le rectangle et donc une dérivée temporelle non nulle du flux magnétique.
Dans le temps « dt », l’onde se déplace d’une distance « c dt ». En se déplaçant, il balaie une aire égale à « ac dt ».
Pendant cet intervalle de temps, le flux magnétique à travers le rectangle augmente, ce qui donne le taux de variation du flux magnétique.
Cette valeur est substituée dans la loi de Faraday, ce qui donne une relation entre E et B en termes de vitesse de propagation des ondes dans le vide.

<ol>
	<li>Young, H.D. et Freedman, R.A. (2012). La physique universitaire avec la physique moderne. San Francisco, Californie : Pearson. L’article 32.2 ; page 1056.</li>
	<li>OpenStax. (2019). Physique universitaire Vol. 2. [Version Web]. Récupéré de <a href="https://openstax.org/details/books/university-physics-volume-2">https://openstax.org/details/books/university-physics-volume-2</a> ; l’article 16.2 ; Pages 710 et 711.</li>
</ol>

ondes électromagnétiques planes ii

Considérez une front d'onde plane se déplaçant dans la direction x avec une vitesse constante. Ce front d'onde peut être utilisé pour obtenir la relation entre les champs électrique et magnétique à l'aide de la loi 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;" />

Pour appliquer la loi de Faraday, considérez un rectangle de largeur a, comme illustré dans la figure 1, dont le vecteur de surface est orienté dans la direction z positive. Pour résoudre l'intégrale du côté gauche de la loi de Faraday, intégrez dans le sens inverse des aiguilles d'une montre le long du rectangle. Le champ électrique est nul le long d'une des longueurs située à l'extérieur du front d'onde, tandis que le champ électrique est perpendiculaire aux deux autres côtés. Ces trois côtés ne contribuent pas à l'intégrale. Seul un côté, le champ électrique a une valeur non nulle et il est parallèle à la longueur. Ce côté contribue à l'intégrale, donnant une valeur non nulle.

<img alt="Equation2" src="/files/ftp_upload/13823/13823_Equation_PT_2.svg" style="text-align: center; height: 35px;" />

Étant donné que le côté gauche de la loi de Faraday est non nul, pour satisfaire l'équation il doit y avoir une composante du champ magnétique dans la direction z positive, qui peut fournir un flux magnétique non nul à travers le rectangle et donc une dérivée temporelle non nulle du flux magnétique. Pour obtenir cette valeur, considérez que pendant un laps de temps dt, l'onde se déplace d'une distance c dt. En se déplaçant, elle balaye une surface égale à ac dt. Pendant cet intervalle de temps, le flux magnétique à travers le rectangle augmente, donnant la vitesse de variation du flux magnétique. Cette valeur est substituée dans la loi de Faraday, résultant en une relation entre E et B en termes de la vitesse de propagation de l'onde dans le vide.

<img alt="Equation3" src="/files/ftp_upload/13823/13823_Equation_PT_3.svg" style="text-align: center; height: 14px;" />

Cette expression montre que l'onde est conforme à la loi de Faraday uniquement si la vitesse de l'onde et les magnitudes des vecteurs perpendiculaires sont liées, comme dans l'équation ci-dessus.

Ondes électromagnétiques planes II

Considérons un front d’onde plan se déplaçant dans la direction x avec une vitesse constante. Ce front d’onde peut être utilisé pour obtenir la relation entre ...

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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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