8.02x - Module 12.01 - EM Plane Waves - Poynting Vector - E-fields - B fields - Wavelength
Summary
TLDRThe video script discusses the fundamental principles of traveling electromagnetic waves, emphasizing the importance of Maxwell's equations and their consequences. It explains how the electric field vector (E) and magnetic field vector (B) are perpendicular to each other and the direction of propagation, always in phase, and how the amplitude of the magnetic field (B0) is related to the electric constant (ε0) and the speed of light (c). The script provides a detailed example of calculating properties of an electromagnetic wave with a frequency of 4.0 * 10^13 Hz, including its wavelength, wave number, and direction of propagation. It also illustrates how to determine the direction of the electric field vector when given the magnetic field vector's oscillation and propagation direction, highlighting the significance of the signs in the cosine function for wave propagation.
Takeaways
- 🧲 Maxwell's Equations: The lecturer emphasizes the importance of Maxwell's equations in understanding traveling electromagnetic waves.
- 🌐 Wave Propagation: The E vector and B vector are always perpendicular to each other and to the direction of wave propagation.
- 🔄 Phase Relationship: E and B are in phase, meaning they reach their maximum and zero values at the same moments in time.
- 🌌 Electromagnetic Wave in Vacuum: The amplitude of the magnetic field (B0) is related to the amplitude of the electric field (E0) through the equation B0 = E0/c.
- 🚀 Energy Flux: The energy per second flowing through each square meter is given by the cross product of E and B vectors divided by the magnetic constant (μ0).
- 🌀 Wave Characteristics: The wavelength (Lambda) is related to the speed of light (c) and the period of oscillation, and the wave number (k) is the reciprocal of the wavelength.
- 🔢 Frequency and Wavelength: Given a frequency of 4.0 * 10^13 Hz, the angular frequency (Omega) and wavelength (Lambda) are calculated, identifying the wave as infrared radiation.
- 📍 Direction of Propagation: The direction of wave propagation is specified as the positive Y direction, with the B vector oscillating in the X direction.
- 📉 Oscillation and Propagation: The B vector's oscillation is described by a cosine function that includes the angular frequency and wave number, crucial for the wave's directionality.
- 🛑 Significance of Signs: The correct signs in the cosine function are essential for the correct direction of wave propagation; changing signs alters the direction.
- 🔄 Changing Propagation Direction: To change the direction of wave propagation, one must adjust the signs in the cosine terms and ensure the direction of E cross B is consistent with the new direction.
Q & A
What are the fundamental properties of electromagnetic waves as described in the script?
-The script describes that the electric field vector (E) is always perpendicular to the magnetic field vector (B), and both are perpendicular to the direction of propagation. Additionally, E and B are always in phase, meaning they reach their maximum and zero values at the same moments in time.
What is the relationship between the amplitude of the magnetic field (B0) and the electric field (E0) in a vacuum?
-In a vacuum, the amplitude of the magnetic field (B0) is related to the electric field (E0) by the equation B0 = E0/c, where c is the speed of light.
What does the pointing vector represent and how is it calculated?
-The pointing vector represents the amount of energy per second flowing through each square meter of a particular surface. It is calculated as the cross product of the electric field vector (E) and the magnetic field vector (B), divided by the permeability of free space (μ0), or E x B / μ0.
What is the significance of the wave number (k) and how is it related to the wavelength (λ)?
-The wave number (k) is a measure of the number of wavelengths per unit length and has units of 1/meter. It is related to the wavelength (λ) by the equation k = 2π/λ.
How is the frequency of electromagnetic radiation related to its wavelength?
-The frequency (f) of electromagnetic radiation is inversely related to its wavelength (λ). This relationship is given by the equation λ = c/f, where c is the speed of light.
What is the significance of the angular frequency (ω) in the context of the script?
-The angular frequency (ω) is related to the frequency of the electromagnetic wave and is used to describe the rate of oscillation. It is calculated as ω = 2πf, where f is the frequency of the wave.
What is the frequency given in the script, and how does it relate to the angular frequency and wavelength?
-The frequency given in the script is 4.0 * 10^13 Hz. The angular frequency (ω) is then 2.5 * 10^14 rad/s, and the wavelength (λ) is 7.5 * 10^-6 m, which corresponds to infrared radiation.
What is the direction of propagation (V) and the oscillation direction of the magnetic field (B) vector as described in the script?
-The direction of propagation (V) is in the positive Y direction. The magnetic field (B) vector oscillates in the X direction, as indicated by the cosine function with a time-dependent term and a spatial term with a wave number k.
Why is the sign in front of the wave number (k) in the expression for the magnetic field (B) vector crucial?
-The sign in front of the wave number (k) is crucial because it determines the direction of wave propagation. A positive sign indicates propagation in the positive Y direction, while a negative sign would indicate propagation in the negative Y direction.
How does the direction of the electric field (E) vector relate to the magnetic field (B) vector and the direction of propagation (V)?
-The electric field (E) vector must be in a direction such that the cross product of E and B is in the direction of propagation (V). In the script's example, E is in the positive Z direction, ensuring that E x B is in the positive Y direction, consistent with the wave propagating in the positive Y direction.
What changes would be necessary to change the direction of propagation from positive Y to negative Y?
-To change the direction of propagation from positive Y to negative Y, the signs within the cosine terms for both E and B would need to be changed to be the same (both positive or both negative), and the direction of the electric field (E) vector would need to be adjusted so that E x B is in the negative Y direction.
Outlines
🌌 Electromagnetic Waves and Maxwell's Equations
The first paragraph introduces the concept of traveling electromagnetic waves, emphasizing the importance of Maxwell's equations in understanding their behavior. The speaker admits to knowing few equations but being able to derive many from those known. Key points include the perpendicularity of the electric (E) and magnetic (B) vectors to the direction of propagation, their in-phase nature, and the formula for the amplitude of the magnetic field in vacuum (B0 = E0 * c). The energy flux, given by the cross product of E and B divided by the magnetic constant (μ0), is also discussed, along with the relationship between wavelength (λ), speed of light (c), and frequency (f). The paragraph concludes with an example problem involving an infrared wave with a given frequency, direction of propagation, and magnetic field oscillation.
📏 Directionality and Propagation of Electromagnetic Waves
The second paragraph delves into the specifics of wave propagation direction, highlighting the crucial role of the sign in the wave equation. It explains how a change in sign affects the direction of wave propagation, using the cosine function to illustrate the relationship between time (t), angular frequency (ω), and wave number (k). The paragraph also discusses the importance of maintaining the right-handed coordinate system and the perpendicular relationship between the electric and magnetic fields. It concludes with a practical example of how to determine the direction of the electric field vector (E) based on the given magnetic field vector (B) and propagation direction, ensuring that E x B aligns with the direction of propagation.
🔄 Changing the Propagation Direction of Electromagnetic Waves
The third paragraph focuses on altering the direction of wave propagation. It explains that changing the signs within the cosine terms is not sufficient; one must also ensure that the cross product of the electric and magnetic fields (E x B) changes direction accordingly. The paragraph provides a step-by-step guide on how to adjust the signs to achieve the desired propagation direction, emphasizing the need to consider the overall orientation of E x B. It concludes with a hypothetical scenario where the wave is made to propagate in the opposite direction, illustrating the necessary sign changes.
Mindmap
Keywords
💡Electromagnetic Wave
💡Maxwell's Equations
💡Perpendicularity
💡Direction of Propagation
💡Amplitude
💡Energy Flux
💡Wavelength
💡Wave Number
💡Angular Frequency
💡Phase
💡Poynting Vector
Highlights
Maxwell's equations are fundamental to deriving properties of traveling electromagnetic waves.
Electromagnetic waves are characterized by the E vector being perpendicular to the B vector.
Both E and B vectors are perpendicular to the direction of wave propagation.
E and B vectors are in phase, reaching maximum and zero values simultaneously.
The amplitude of the magnetic field (B0) is related to the speed of light in a vacuum.
The energy flux is given by the cross product of E and B vectors divided by the permeability of free space (μ0).
The wavelength (λ) is related to the speed of light (c) and the period of oscillation.
Introduction of wave number (k) as a unit of 1/meter, derived from the wavelength.
The angular frequency (ω) is directly proportional to the wave number (k) and the speed of light.
A frequency of 4.0 * 10^13 Hz is given, leading to specific calculations for ω and λ.
The direction of wave propagation is specified as the positive Y direction.
The B vector oscillates in the X direction with a given angular velocity (ω).
The importance of the correct sign in the wave equation for proper propagation direction.
The relationship between the E and B vectors and their phase synchronization.
The E vector is determined to be in the positive Z direction based on the right-hand rule.
A step-by-step explanation of how to determine the direction of the E vector in relation to B and propagation direction.
Changing the direction of wave propagation requires adjusting both the signs in the wave equation and the direction of E cross B.
Transcripts
problem one is
335 it deals with a
traveling electromagnetic
wave now I told you earlier and I wasn't
joking that I know very few equations in
physics I can derive many from the few
that I know I know Maxwell's equation s
yes I do certainly after this
course I also know some of the
consequences of Maxwell's equations
related to traveling waves and I will
share those with you and if you remember
those because it would be rather silly
to derive them every time then it's very
easy to write down traveling
electromagnetic waves with the vectors
in the right direction and the
amplitudes all correct and the plus and
minus signs correct the first thing that
I happen to remember is that the E
Vector is always perpendicular to the B
Vector the second is that e as well as
B are both perpendicular to the
direction of
propagation the third is that e and B
both are in face at all moments in time
what does that mean it means if one
reaches a maximum value the other
reaches a maximum value at the same
moment in time of course at right angles
and if one goes through zero the other
goes through zero that's what it means
always in
phase and then B 0 which is the
amplitude of the magnetic field Factor
equals e0 ided c when the electr
magnetic radiation is in
vacuum
now independent of electromagnetic waves
what always holds is that the pointing
Vector which tells you how much energy
per second flows per square
meter out of a particular surface equals
e cross B divided by mu0 so this is
energy per second flowing through each
square meter and I will call that flux
it's an energy
flux what also so always
holds Lambda equals c * the period of
one oscillation don't confuse this T's
Tesla so this is also C divided by the
frequency of the radiation
F what you will often see introduced is
a wave number K which has nothing to do
with the unit Vector in the Z Direction
nothing that k equals 2 Pi / Lambda and
that has as a unit 1 over meter now
since Omega equals 2 pi * the frequency
f it follows immediately that Omega
divided by K also is the speed of light
the speed with which electromagnetic
radiation propagates itself in vacuum
now if you can remember all these things
that would help I happen to remember
them we have here in problem one we have
a frequency which is
4.0 * 10 13
Hertz so it follows immediately that
Omega equals
2.5 * 10
14th radians per
second and it follows that Lambda equals
7.5 * 10- 6 M 75,000 angstroms if you
like that unit this is infrared
radiation we're also being told that the
direction of V is in plus
Y and we're being told that the B Vector
oscillates in the direction of X plusus
plusus plus minus with that angular
velocity
Omega and you know b0 whatever that is
so the B vector
if we write that down in terms of X Y
and Z and all moments in time would be
the amplitude of the B Vector times
cosine Omega T which indicates the
oscillatory character Min - k
y in the X
Direction and K then here mean simply 2
Pi / Lambda if you want to write for
this k 2 Pi / by Lambda be my guest now
let's look at these
things this minus sign is absolutely
crucial If This Were A plus sign the
wave would not be propagating in the
plus y direction and you can immediately
check that for yourself increase T by a
teeny weeny little bit if you want the
cosine function to be exactly the
same you would have to increase y by a
teeny weeny little bit so that Omega T
minus KY Remains the
Same so for this wave to propagate in
the plus y direction it is important
that this sign here and this sign here
have an opposite sign if they both have
the same sign minus minus or plus plus
then the wave would run in the minus y
direction and this X tells you simply
that the B Vector oscillates in the X
Direction sometimes plus sometimes
minus well Omega you already know if you
are interested in K by any chance then I
believe that is
8.38 * 10 5 m minus
1 now since you know b0 you also know e0
because e0 equals B 0 *
C so if we write down now the E Vector
as a function of XYZ and T then we would
get e z here which you
know and then I get exactly this
same argument Omega T minus
KY why because I mentioned earlier that
e and B are always at any moment in time
exact in PH and that's only possible if
this is identical to
that now comes the question what is the
direction that I should put here what is
the unit Vector that I should put here
and for that I'm going to make a
drawing and I'm going to make a drawing
of a right-handed coordinate system
always make a drawing of a right-handed
coordinate system and a right-handed
coordinate
system is a coordinate system whereby X
cross y equals z don't even think of
ever doing it in another way because you
get into deep
trouble so let this be
X and let this be
y then
this would be Z convince yourself that X
cross y would then be
Z now pick any moment in time and let us
assume that the B Vector is in this
direction of course it's not only along
the X Direction in this direction but
since it's a Plaine wave it's everywhere
here the same magnitude and the same
direction and this wave propagates to
the
right and a little later in time half a
period in time if assume that this is
the maximum value B possible in this
direction
half a period in time it will be
pointing down and a quarter period in
time it will go through zero and they
will all go through zero at exactly the
same
time in what direction at this moment in
time should I now Point
e so that and this is one of
my statements that I made earlier so
that e cross B is in the direction of V
which means in this case in the
direction of positive y
well that should not give you any
problems the only way that you can do
that is when e is in the
plus Z Direction convince yourself that
only then is e cross B in the plus y
direction so that means that if we now
have to finish this Vector notation for
E we have to write down here a z roof
and it's all done I canot asked the
question suppose we wanted to move that
wave in the other direction not in plus
y but in minus y what would we have to
change well we could change this minus
sign to a plus sign and this minus sign
to a plus sign what count is actually is
not so much that this becomes a plus but
that these two signs here and here are
the
same but that's not enough if you want e
cross B to be in the opposite direction
you also must either put a minus sign
here so you get a minus X and leave the
plus here or you put a minus sign here
in the minus Z Direction and leave the
plus there so whenever you have to
change the direction of propagation keep
in mind that it's not enough to only
change the signs inside the brackets of
the cosine terms but you also have to
take into account that e cross B must
change direction
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