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Chapter 3.1a Waves

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the topic of this video is

electromagnetic energy and specifically

waves the learning objectives are on the

screen so you can go ahead and pause

now to write these down in your notes

i'm going to

jump right into talking about basic

behavior of waves

and describing the difference between a

traveling wave and a standing wave

so to to start let's take a look at a

traveling wave

on the screen is a a graphical

representation of a traveling wave

and if if you uh are in any doubt of

of identifying a traveling wave all you

need to do is pick

a point on the wave and follow its

motion so let's pick a

peak which is the highest point of the

wave and follow its motion

and you see that it the wave is moving

from left to right or traveling

in in a particular direction you can

pick a different peak

such as the trough which is the lowest

point also follow its

net motion as the wave uh uh is

moving by here we see that it is indeed

traveling so this is an example of a

traveling wave

to contrast the traveling wave i have

here standing wave

so a standing wave uh as opposed to a

traveling wave

um you can see if you pick a particular

point it is

not moving like a traveling wave uh

uh wood instead the the peak here

is just inverting to become a trough and

reinverting back to become

a peak and so on and so forth and this

is actually happening in multiple segments

segments

of this particular standing wave one of

the key aspects of the standing wave and

i'll refer

resort back to this at the end of the

video is this uh

red circle here in between these areas

of peak to trough inversions

these are called nodes and this is where

it's a part of the wave that's not

there's no net movement at all

despite uh peak to trough inversions

occurring on either side of it

okay now the electromagnetic

nature of light has to do with a

traveling wave

so let's go back to the traveling wave

where uh the wave

is is moving we can follow any point on

the wave and it is moving through space

um light let's see if i can zoom in here

if that is beneficial

is a traveling wave that is occurring in the

the

uh electromagnetic field

and the electromagnetic field is

essentially uh

perpendicular oscillations in the in

electric field and a magnetic field okay

so that's why we call this

electromagnetic radiation

so light can be thought of as a

traveling wave

okay and um and and

there are some fundamental uh aspects of

light as a traveling wave that we need

to consider okay but

when we talk think about light as a wave

and particularly as a traveling wave

um this is what we're referring to as

electromagnetic radiation

uh in oscillation a perpendicular

oscillation in

an electric field and the perpendicular

so to help us describe waves more generally

generally

we can look at this figure that i'm

showing on the screen right now

and in this figure there are a few uh

fundamental aspects of waves that are

highlighted for us the first is the wavelength

wavelength

which is you uh we denote the wavelength

using the symbol

the greek symbol lambda lowercase lambda

and that is going to be the distance in

meters between two equivalent points in

a wave so here we can pick two peaks

that are consecutive and the distance

between those two consecutive peaks is

the wavelength

um the another fundamental aspect of a

wave is the frequency

denoted with the greek letter nu uh

which is sort of like a v with a with a um

sort of like a v and a u hybrid letter um

um [Music]

[Music]

and this is going to refer to the number

of cycles per second

uh of a wave that is how many um for

example how many peaks pass through a given

given

point in space in a given amount of time

so here we can see that this for this

first wave that we're looking at

with a relatively large wavelength where

we can count up three cycles per second

um three cycles per second or one

over seconds is going to be also

equivalent to hertz that unit one over

seconds is also

equivalent to hertz okay that's really

important to know

um if we look at what happens now if we

decrease the wavelength

of a wave and look at what happens to

the frequency

over the same distance traveled here in

one second is what i'm looking at

at up top here so that's sort of

bracketed by these

dashed lines on either side that's the

distance traveled in one second

what we see is that we decrease the

wavelength but the frequency

increased if we decrease the wavelength

yet again you can visually see that

there are more

cycles or more peaks that can pass

through this

distance traveled in one second for a

shorter wavelength higher frequency

so there's it seems to be just from a

qualitative comparison

um of these two variables an inverse relationship

relationship

when one goes up the other goes down and

another another aspect of waves is the

amplitude so if you draw an imaginary line

line

that cuts halfway between the peaks and

the trops

between the midway imaginary line and

the peak

or the midway imaginary line and the trough

trough

is the amplitude so you can notice here

that in the top there's an example of a

higher amplitude wave

on the bottom a lower amplitude wave but

the wavelength

that is the peak to peak separation is

the same here

and the frequency the number of peaks

second or cycles per second

is the same so amplitude is independent of

frequency and wavelength

i'm going to go ahead and write down

some definitions here for wavelength the

wavelength is

the distance between two consecutive

peaks or troughs in a

the number of wave cycles

that pass a specified

point in space in a

specified amount of time

magnitude

of the waves displacement

okay so now what we can do is i

mentioned this inverse relationship between

wavelength and frequency and so

this is actually uh really interesting

because if you multiply the two together

if you take wavelength and you multiply

it by frequency think about the units

right where meters um is is the si unit

of distance

for a wavelength and um frequency is

going to be

uh hertz or you know reciprocal seconds

times one over seconds

what we get is a unit value of meters

per second

which is a speed and it turns out that

for electromagnetic radiation

not just any waves in general but for electromagnetic

electromagnetic

radiation the multiplication of

the wavelength and the frequency will

give actually a fixed

speed so if we take

for electromagnetic radiation wavelength

times frequency

we use the the the

variable c and this is actually the

speed of light

and so this is a fundamental constant of

the value 2.998

times 10 to the 8th meters per second

we could have more significant figures

but this 4 sig figs

is is going to be good for most purposes

okay so why don't we go ahead and apply

this uh

right away to a practice problem um so

here we're going to be determining the

frequency and wavelength

of radiation so you can pause the video now

now

and work on this problem uh and then

uh resume the video once so you can compare

compare

so what i'm going to do here is i'm

going to start by

revisiting the calculation or the the

equation that i just wrote so wavelength

times frequency

is equal to c the speed of light we are given

given

up here um a wavelength

and we're given a conversion factor so

and we know that the speed of light is a

constant so what we need to do is solve for

for

or isolate frequency so frequency

is going to be equal to the speed of light

light

c over uh lambda the wavelength

so uh one way to do this is to just go

ahead and

plug in what we have 2.998 times

10 to the 8th meters per

second i'm actually going to change my notation

notation

quickly so the speed of light

is given in meters per second that's the

units um

you can also write this as meters times

uh reciprocal seconds okay so that that

is really useful

um sometimes for dimensional analysis

problems keeping track of what's a

numerator unit what's a denominator unit

so that's the the speed of light divided

by the wavelength of 589

nanometers now personally uh

i would convert this nanometer value to

meters first before putting this in but

i want to show you that you can actually

handle this

um in in using a dimensional analysis type

type

of approach but keep in mind that the

nanometers down here will not cancel out

with the meters up on top

because they're not the same unit we

actually have to

uh fix this problem so

to cancel nanometers on the bottom over

here i'm going to write nanometers up on

the top to cancel

meters on the numerator to the left i'm

going to write meters

in the denominator here

so the conversion factor we could do a

couple different things but i'll stick with

with

1 meter is equal to

actually let's use the conversion factor

given to us by um the problem here

so we have um one nanometer is equal to

one times ten to the negative nine meter

let's just use that conversion

so now what we can do is let's always

check that our units cancel out

meters will cancel out with meters over

here nanometers cancel out with

nanometers over here

okay so this is the dimensional analysis

part we treat the units

as if they are being modified by

multiplication and division

uh okay so if you do this what you

should get

is a value of 5.09

times 10 to the 14th

per second per second is the only unit left

left

okay so that's that's a frequency so

does this answer make sense yes

it does all right to wrap up this video

i really

briefly just want to discuss quantization

quantization

and the definition is on your screen it

is where only discrete values from a

more general set of continuous values of

some property

are observed okay what does that mean

so let's take the case of a standing

wave these black circles on the left and

right are fixed points they are not

moving to

to jostle the the weight if one of them

was moving to jostle the wave then you

might get a traveling wave where

a peak moves from left to right but

since they're both locked there

and let's say some other energy source

is being used to

to um uh move this rope

uh to provide a standing wave well what

happens is we can only have certain frequencies

frequencies

in this particular case for a standing

wave why is that the case

well we can see here that

in the most simplest wave form there is no

node right there is a peak

and a trough and it is simply just

inverting between those two

okay with fixed points on the left and

the right

so we can say here that the node

is zero but if we want to

jump up to uh one node now in the next case

case

okay if we want to actually increase the

frequency of this standing wave

to get it to one node there's only uh

we can't pick any frequency that's

higher than the

than the first waveform we actually have

to pick

a very discrete frequency to get up to

one node okay so for one node

two nodes or three nodes these correspond

correspond

to discrete frequencies to get to these

points okay

so um let's just say

to discrete

for each i'll call it

a wave form okay that's just a different

type of standing wave that we have

so each waveform here corresponds to a

discrete frequency

that is an example of quantization

because in theory

you know for traveling waves we can we

can have

um a huge spectrum of frequencies we can

almost have you have as many frequencies

as you could possibly

think of for electromagnetic radiation

um the only thing that will happen with

the frequency is that the wavelength

will change inversely

but for standing waves where the ends

are fixed

that is the case where only certain

frequencies will enable that waveform to exist

exist

this is quantization and this is going

to be a really

key part of understanding the electronic

structure of matter

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YouTube Transcript:
Chapter 3.1a Waves