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

K12 Standards

Virginia  Standards of Learning

HIgh School

Physical Science

PS.7

PS.7.a

Plane Electromagnetic Waves II

電場と磁場の関係は、ファラデーの法則を+x方向に一定の速度で移動する平面の波面に適用することによって得られます。
+z 方向の面積ベクトルを持つ高さ "a" の四角形を考えてみましょう。
長方形の周りを反時計回りに積分すると、電界は長さ「PQ」に沿ってゼロになるか、長さ「RS」以外の要素に対して垂直であることがわかります。これは、ゼロ以外の値を与える積分にのみ寄与します。
ファラデーの法則を満たすには、+z方向に磁場成分が必要であり、これにより長方形を介して非ゼロの磁束、したがって磁束の非ゼロ時間微分を提供できます。
時間 "dt" では、波は "c dt" の距離を移動します。移動中は「acdt」に等しい面積を掃引します。
この時間間隔では、長方形を通る磁束が増加し、磁束の変化率が得られます。
この値はファラデーの法則に代入され、真空中の波の伝播速度に関してEとBの間に関係が生じます。

<ol>
	<li>ヤング、H.D.およびフリードマン、RA(2012)。現代物理学と大学物理学。カリフォルニア州サンフランシスコ:ピアソン。セクション32.2;1056ページ。</li>
	<li>OpenStaxです。(2019). 大学物理学 Vol.2.【Web版】。<a href="https://openstax.org/details/books/university-physics-volume-2">https://openstax.org/details/books/university-physics-volume-2 から取得</a>。セクション16.2;710〜711ページ。</li>
</ol>

平面電磁波 ii

位置x方向に一定の速度で進む平面波面を考えてみましょう。 この波面を利用して、ファラデーの法則を利用して電場と磁場の関係を取得できます。

<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;" />

ファラデーの法則を適用するには、図 1 に示すように、面積ベクトルが正の z 方向にある幅 a の長方形を考えます。 ファラデーの法則の左側積分を解くには、長方形に沿って反時計回りに積分します。 波面の外側にある長さの 1 つに沿って電場はゼロですが、電場は他の 2 つの長さ要素に対して垂直です。 これら 3 つの辺は積分に寄与しません。 電界は片側に沿ってのみゼロ以外の値を持ち、長さに平行です。 この側は積分に寄与し、ゼロ以外の値が得られます。

<img alt="Equation2" src="/files/ftp_upload/13823/13823_Equation_PT_2.svg" style="text-align: center; height: 35px;" />

ファラデーの法則の左辺はゼロではないため、方程式を満たすには、正のz方向に磁場成分が存在する必要があります。 長方形を通るゼロ磁束、したがって磁束の非ゼロ時間導関数。 この値を取得するには、 時間dt内に波が距離c dt移動することを考慮します。 移動中はac dtに等しいエリアをスイープします。 この時間間隔中に、長方形を通る磁束が増加し、磁束の変化率が得られます。 この値はファラデーの法則に代入され、真空中の波の伝播速度に関する E と B の関係が得られます。

<img alt="Equation3" src="/files/ftp_upload/13823/13823_Equation_PT_3.svg" style="text-align: center; height: 14px;" />

この式は、上の方程式のように、波の速度と垂直ベクトルの大きさが関連している場合にのみ、波がファラデーの法則と一致することを示しています。

平面電磁波 II

平面の波面が一定の速度で x 方向の位置に移動するとします。この波面を利用して、...

Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Electromagnetic Waves

電磁波

KEY TERMS AND CONCEPTS

主要用語と概念

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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key terms and concepts

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

physics

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Minnesota Academic Standards in Science

9-12

3.2.2 9P.3.1.1.2

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New Jersey Student Learning Standards for Science

High School

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HS-PS4-5

Alabama Course of Study Science

9-12.P.3.7.c

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HS-PS4-4.

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PS.7.b

Faraday’s Law in a Plane Wave

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