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

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 $x$ direction with its electric field $\vec{E}$, of amplitude $300 \mathrm{~V} / \mathrm{m}$, directed along the $y$ axis. (a) What is the frequency $f$ of the wave? (b) What are the direction and amplitude of the magnetic field associated with the wave? (c) What are the values of $k$ and $\omega$ if $\vec{E}=\vec{E}^{\max } \sin (k x-\omega t) ?$ (d) What is the time-averaged rate of energy flow in watts per square meter associated with this wave? (e) If the wave falls on a perfectly absorbing sheet of area $2.0 \mathrm{~m}^{2}$, at what rate is momentum delivered to the sheet and what is the radiation pressure exerted on the sheet?

A plane electromagnetic wave, with wavelength $3.0 \mathrm{~m}$, travels in vacuum in the positive $x$ direction with its electric field $\vec{E}$, of amplitude $300 \mathrm{~V} / \mathrm{m}$, directed along the $y$ axis.
(a) What is the frequency $f$ of the wave? (b) What are the direction and amplitude of the magnetic field associated with the wave? (c) What are the values of $k$ and $\omega$ if $\vec{E}=\vec{E}^{\max } \sin (k x-\omega t) ?$ (d) What is the time-averaged rate of energy flow in watts per square meter associated with this wave? (e) If the wave falls on a perfectly absorbing sheet of area $2.0 \mathrm{~m}^{2}$, at what rate is momentum delivered to the sheet and what is the radiation pressure exerted on the sheet?

Key Concepts

Momentum Transfer and Radiation Pressure

Electromagnetic waves carry momentum in addition to energy, and when they interact with a surface, they can exert pressure known as radiation pressure. This pressure results from the momentum transfer during absorption or reflection, and is quantitatively related to the energy flux of the wave. Understanding momentum transfer is critical in contexts such as light sails, solar radiation pressure, and the interaction of light with matter.

Poynting Vector and Energy Flow

The Poynting vector represents the rate of energy transfer per unit area in an electromagnetic field, defined as S = (E x B)/?0. For time-averaged energy flow calculations in oscillating fields, this concept is used to determine the intensity of the wave, which is an important quantity in understanding how energy is transmitted through space.

Wave Number (k) and Angular Frequency (?)

The wave number k and angular frequency ? are measures of the spatial and temporal periodicity of a wave, respectively. They are defined as k = 2?/? and ? = 2?f. These parameters are essential in describing the phase of the wave, as seen in expressions like E = E_max sin(kx - ?t), indicating how the wave oscillates in both space and time.

Relationship Between Wavelength, Frequency, and Speed of Light

The wavelength and frequency of an electromagnetic wave are directly related to its speed through the equation c = f?, where c is the speed of light in vacuum. This relationship allows one to compute the frequency when the wavelength is known, or vice versa, and is fundamental to understanding the dispersion characteristics of waves.

Electromagnetic Waves

Electromagnetic waves are solutions of Maxwell's equations in free space, characterized by oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation. They are transverse waves that transport energy, momentum, and can exhibit both particle and wave-like behavior.

Electric and Magnetic Field Relationships

In a plane electromagnetic wave, the electric field and magnetic field are perpendicular to each other and to the direction of propagation. The magnetic field's amplitude is related to that of the electric field by the speed of light, specifically B = E/c in vacuum. This relation arises from Maxwell's equations and ensures the self-sustaining propagation of the fields.

Transcript

00:01 So in this question we're asked all about an electric field, which is traveling in the positive x direction and oscillating in the along the y axis.

00:10 And it has a wavelength of three meters and a maximum amplitude of 300 volts per meter.

00:17 Then we're asked numerous questions about this electric field.

00:20 So we're asked first, what is the frequency of the wave? then what is the direction and amplitude of the magnetic field associated with the wave? then what is the wave number and what is the angular frequency of the wave then what is the time average rate of energy flow in watts per meter squared of the wave and then finally if the if the wave falls perfectly on a on a sheet that perfectly absorbs light of area two two meters squared what is the rate of momentum delivered to the sheet and the rate what is the radiation pressure exerted on the sheet.

01:00 So i've written down everything that we're getting.

01:01 So we're given the magnitude is 300 volts per meter.

01:04 The wavelength is 3 meters.

01:05 Speed of light is 3 by 10 to the 8 meters per second.

01:10 The area of the sheet is 2 meters squared.

01:16 And we know that the electric field is of this form.

01:20 So it's of the vector e is equal to the amplitude e0 by sign of kx, where k is the wave number minus omega t, where omega is the angular frequency and we're told that it's in the it's traveling in the positive x direction so then we have the i -hat vector here or unit vector here so for part a to calculate the frequency to start off by calculating the frequency so we know that for any way that the speed of the wave v which is in this case the speed of light c is equal to the frequency by the wavelength so that means that the wavelength is equal to this speed of light divided by the frequency this means that the frequency is equal to the speed of light divided by the wavelength.

02:07 So this is equal to 3 by 10 to the 8 over 3 meters, which is equal to 1 by 10 to the 8 hertz.

02:18 So that's our answer for part 8.

02:19 For part b, we're asked what is the direction and the magnitude of the magnetic field associated with the field? so we know the relationship between the magnitude of the electric and the, the magnitude of the electric and the and the magnetic field is given by e, the magnitude of the electric field over the magnetic field is equal to the speed of light so that implies that our might weather the magnetic field that we're trying to find is equal to the magnitude of the electric field over the speed of light which is equal to 300 volts per meter over 3 right 10 to the 8 meters per second.

03:00 If we work this out this is equal to then is equal to 1 .1 by 10 to the minus 6 tesla.

03:14 And of course, then the direction is and it's in the same direction as e field.

03:22 Since they propagate in the same direction, same direction as the e field.

03:29 So that means it's in the positive x direction.

03:40 So these are answers for part a and b.

03:50 So for part c, we're asked, we're asked at what is the wave number and angular frequency.

03:57 So we know the wave number of the wave is, of any waves, equal 2 pi over the wavelength.

04:03 So this is equal to 2 pi over the lambda, which is equal to 3 meters.

04:07 And if we work this out, this is equal to 2 .09 watts per meter squared...

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