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

The relation between electric and magnetic fields is obtained by applying Faraday&#39;s law to a plane wavefront traveling in the +x-direction with a constant speed.
Consider a rectangle of height &#34;a&#34; having an area vector in the +z-direction.
While integrating around the rectangle counterclockwise, the electric field is found to be zero along length &#34;PQ&#34;, or perpendicular to the elements except for length &#34;RS&#34;. It only contributes to the integral giving a non-zero value.
To satisfy Faraday&#39;s law, there must be a magnetic field component in the +z-direction, which can provide a non-zero magnetic flux through the rectangle and hence a non-zero time derivative of magnetic flux.
In time &#34;dt&#34;, the wave moves a distance &#34;c dt&#34;. While moving, it sweeps an area equal to &#34;ac dt&#34;.
During this time interval, the magnetic flux through the rectangle increases, giving the rate of change of magnetic flux.
This value is substituted in Faraday&#39;s law, resulting in a relationship between E and B in terms of the speed of wave propagation in a vacuum.

plane electromagnetic waves ii

Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between electric and magnetic fields with the help of Faraday's law.
<img alt="Faraday's law equation, &#8750;E&#183;dl=-d&#934;B/dt, involves electromagnetic induction principles." src="/files/ftp_upload/13823/13823_Equation_PT_1.svg" style="text-align: center; height: 35px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
<img alt="3D vector field diagram illustrating force vectors and planes in static equilibrium analysis." src="/files/ftp_upload/13823/13823_PT%20_Figure_1.svg" style="text-align: center; height: 175px;" data-altupdatedat="1761830439000" data-alt="Figure1" />
To apply Faraday's law, consider a rectangle of width a, as shown in the figure.1, whose area vector is in the positive z-direction. To solve the left-hand side integral in Faraday's law, integrate counterclockwise along the rectangle. The electric field is zero along one of the lengths lying outside the wavefront, while the electric field is perpendicular to the other two length elements. These three sides give no contribution to the integral. Only along one side, the electric field has a non-zero value, and it is parallel to the length. This side contributes to the integral, giving a non-zero value.
<img alt="Electric field line integral equation; Maxwell's law; closed loop integral; electromagnetic concept." src="/files/ftp_upload/13823/13823_Equation_PT_2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
Since the left side of Faraday's law is non-zero, to satisfy the equation there must be a magnetic field component in the positive z-direction, which can provide a non-zero magnetic flux through the rectangle and hence a non-zero time derivative of magnetic flux. To obtain this value, consider that in time dt, the wave moves a distance c dt. While moving, it sweeps an area equal to ac dt. During this time interval, the magnetic flux through the rectangle increases, giving the rate of change of magnetic flux. This value is substituted in Faraday's law, resulting in a relationship between E and B in terms of the speed of wave propagation in a vacuum.
<img alt="Static equilibrium formula E=Bc; mathematical equation; educational diagram." src="/files/ftp_upload/13823/13823_Equation_PT_3.svg" style="text-align: center; height: 14px;" data-altupdatedat="1761830439000" data-alt="Equation3" />
This expression shows that the wave is consistent with Faraday's law only if the wave speed and the magnitudes of the perpendicular vectors are related, as in the above equation.

Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between ...

Education

JoVE Core

Physics

Unit 1

Electromagnetic Waves

KEY TERMS AND CONCEPTS

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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