A plane electromagnetic wave, with wavelength 3.0 m, travels in vacuum in the positive directio

A plane electromagnetic wave, with wavelength 3.0 m,...

A plane electromagnetic wave, with wavelength $3.0 \mathrm{~m}$, travels in vacuum in the positive direction of an $x$ axis. The electric field, of amplitude $300 \mathrm{~V} / \mathrm{m},$ oscillates parallel to the $y$ axis. What are the (a) frequency, (b) angular frequency, and (c) angular wave number of the wave? (d) What is the amplitude of the magnetic field component? (e) Parallel to which axis does the magnetic field oscillate? (f) What is the timeaveraged rate of energy flow in watts per square meter associated with this wave? The wave uniformly illuminates a surface of area $2.0 \mathrm{~m}^{2} .$ If the surface totally absorbs the wave, what are $(\mathrm{g})$ the rate at which momentum is transferred to the surface and (h) the radiation pressure on the surface?

Key Concepts

Electromagnetic Plane Waves

Electromagnetic plane waves are solutions to Maxwell's equations in free space where the electric and magnetic fields oscillate sinusoidally and propagate in a fixed direction. These fields are oriented perpendicular to each other and to the direction of propagation, and their amplitudes and phases are intricately linked, making them a cornerstone for understanding classical electromagnetism.

Wavelength and Frequency Relationship

The wavelength of a wave is the distance over which its shape repeats, and it is inversely related to the frequency, which is the number of oscillations per unit time. In vacuum, the relationship between wavelength (?), frequency (f), and the speed of light (c) is expressed by the equation c = ? f. This concept is fundamental in linking the spatial and temporal characteristics of waves.

Angular Frequency and Wave Number

Angular frequency (?) is defined as 2? times the frequency of a wave, giving a measure of how quickly the oscillations occur in radians per second, while the wave number (k) is defined as 2? divided by the wavelength, representing the spatial frequency in radians per meter. These quantities are essential in describing the oscillatory nature of waves in both time and space, and they appear naturally in the mathematical representation of wave phenomena.

Relationship Between Electric and Magnetic Fields

In electromagnetic waves, the amplitude of the magnetic field is directly related to the amplitude of the electric field by the speed of light, typically expressed as E = cB in free space. This relationship reflects the intrinsic balance between the two fields in propagating energy and momentum, and it is a fundamental result derived from Maxwell's equations.

Poynting Vector and Energy Flow

The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field and is given by the cross product of the electric and magnetic fields. The time-averaged value of the Poynting vector provides the average power per unit area transmitted by the wave, which is a central concept in analyzing energy transport in electromagnetic systems.

Momentum Transfer and Radiation Pressure

Electromagnetic waves carry momentum as well as energy, and when such a wave is incident on a surface, momentum can be transferred, exerting a force over that surface. The radiation pressure is the pressure exerted by the wave, often calculated as the intensity of the wave divided by the speed of light for an absorbing surface, linking the concepts of energy flow, momentum, and force in electromagnetic interactions.

Transcript

00:01 For this problem on the topic of electromagnetic waves, we are told that a plain wave with the wavelength of 3 meters is traveling in a vacuum in the positive direction of the x -axis.

00:10 The electric field of amplitude 300 volts per meter is oscillating parallel to the y -axis, and we want to find the frequency, angular frequency, and angular wave number of the wave, the amplitude of the magnetic field component, the axis that is parallel to the magnetic field oscillations.

00:28 We then want to find the time average rate of energy flow in watts per square meter associated with this wave.

00:35 And we are then told that the wave uniformly illuminates the surface of 2 meters square in area.

00:41 And if the surface totally absorbs the wave, we want to know the rate which momentum is transferred to the surface, as well as the radiation pressure that is received by the surface.

00:52 Now, with the wavelength lambda of 3 meters, the frequency from the wave equation, is c over lambda, and this is 2 .998 times 10 to the power 8 meters per second, which is the speed of light and vacuum, over the wavelength of this electromagnetic wave 3 meters, which gives a frequency of 1 times 10 to the power 8 hertz.

01:30 For part b, from the value of the frequency, which we obtained, we can find the angular frequency omega, which is defined to be 2 pi f, and this is 2 pi times the frequency calculated above 1 times 10 to the power 8 hertz, which gives 6 .3 times 10 to the power 8 radiance per second.

01:59 For part c, the angular wave number k is defined as 2 pi over lambda, and this is 2 pi over the wavelength, which we know is 3 meters, which gives the angular wave number to be 2 .1 radiance per meter.

02:34 For part d, we have the electric field amplitude em given to be 300 volts per meter.

02:43 And so the magnetic field amplitude bm is equal to the electric field amplitude over c.

02:52 And so this is 300 volts per meter divided by 2 .998 times 10 to the power 8 meters per second.

03:05 And so we get the magnetic field amplitude to be 1 times 10 to the power minus 6 tesla's...

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