Free Physics NEET Video Lectures for Electromagnetic Waves

00:>00:00 e [Music] a very good morning to all of you so today we have come to the last section of electricity and magnetism and that is electromagnetic waves so today what I will do is I will discuss about electromagnetic waves what do they represent and what are these waves and what kind of frequencies etc etc so recall we have been discussing the basic equations which describe electricity and magnetism so let me write the four equations that we have obtained through the course on the lectures on electricity and magnetism

00:01:22 and these are called maxos equations because Maxwell added a term which I described last time as the displacement current and and that is a very very important contribution that predicted the existence of waves which are now called electromagnetic waves so let me first write down the max equations so first of these equations is integral e. da charge enclosed by 0 integral b. da is equal to 0 integral e do DL is equal to minus d by DT of 5 b which is equal to minus d by DT of integral b. da and finally

00:02:28 integral b. DL is equal to mu0 i+ Mu 0 epsilon0 integral 0 d by DT of integral e d the four equations um in form of integrals and these are the four maxal equations this is nothing but Gus's law we have used uh right in the beginning this tells me that the flux of Electro elect electric field must be equal to the charge enclosed byepson 0 so let me draw a figure here so if you have if you take a surface like this then the electric flux which is coming out of this this are the E Fields this is

00:03:20 integral e. da is equal to charge and Clos by so all it says is that the first question says that the net electric flux coming out from a closed surface this integral with the closed surface must be equal to the charge enclosed divided by epsilon0 so depending on the sign of the charge the electric flux is pointing away from this surface or towards the surface so the charge is positive then we know the electric flux is coming out if charge is negative the electric field lines are entering the volume entering

00:03:55 the surface area also note that as we have discussed before uh if the net charge inside is zero the flux is zero and that does not imply the charges are zero I could have electric Fields with net flux being zero the second equation which is in magnetic field if I take another surface here then I find that integral b. DL b. da must be equal to zero no net no net flux coming coming out because magnetic field lines are Clos lines as many field lines enter the volume as are coming out so magnetic field lines are closed lines and so it

00:04:44 also implies that there are no magnetic charges that means no points from which magnetic field lines emerge or get converged in in other words we say that magnetic monopoles do not exist and so magnetic field lines as we have seen before are always closed lines and so if you take a closed surface the net magnetic flux through that surface will be zero so this is the second law which we written the third law which is farad is law so if you take a a loop and if you have magnetic field lines crossing that

00:05:26 Loop then this equation says integral e. DL is equal to minus d f b by DT that means a changing magnetic flux through this Loop will induce an EMF in the loop or a changing magnetic field will induce an electric field and the direction of the induced EMF is decided by lens's law because of the negative sign the induced EMF is all always to oppose the changes in flux that is farad law of induction and finally we had the fourth equation which was essentially if I draw another figure here and I

00:06:14 had electric field lines crossing this integral b. DL is equal to mu0 i+ Mu 0 epon 0 d by DT of P so magnetic fields get induced either by currents so that means currents can generate magnetic fields or changing electric flux can also generate magnetic fields this is the displacement current which we have discussed before this term is the displacement current which was introduced by Maxwell and it's a very very important modification of ampers law remember we had initially discussed this part which

00:07:08 was ampers law to calculate the magnetic fields produced by currents and what he showed was that to be consistent you have to have another term in this equation which we call as the displacement current so these four laws of Maxwell Maxwell's equations describe all electromagnetism and and they are a beautiful representation of the electrical and magnetic properties of the fields the materials Etc now in free space if I look at a free space for example when there are no charges and no currents these equations

00:07:46 become the following in free space this equation become integral e Doda is equal to0 integral b. da is equal to Zer integral e. DL is equal to minus d by DT of integral b. da and integral B do DL will be equal to Mu 0 epon 0 d by DT of integral e do da so these equations only contain electric and magnetic fields in free space in the absence of any charges or in the absence of currents flowing charges I have these four equations and these four equations actually if you look at these two equations they couple electric and

00:08:49 magnetic fields this electric field depends on magnetic field and magnetic field depends on electric field so these two equations actually couple the electric and magnetic fields and this is what as we will see later on re results in the existence of new forms of waves which are called electromagnetic waves so these are the four maxos equations in uh what are called as integral form in an advanced course when you when you in your career when you go into an advanced course you will see that these equations convert can be

00:09:22 converted into differential equations and they form the four fundamental equations of electromagnetism so what Maxwell did was to generalize ampers law and introduce the displacement current to be consistent and from here he actually derived a wave equation an equation which describes the existence of waves and he found that those waves which are waves of electric and magnetic fields have a certain velocity and the velocity of those waves was approximately 3 into 108 m/ second so this velocity which he calculated was

00:10:03 so close to the measured velocity of light at that time that he propounded he boldly suggested that light waves are electromagnetic waves till that time light waves were not considered to be electromagnetic but he showed that from these equations You can predict the existence of waves which are called electromagnetic waves and the velocity of those electromagnetic waves we found out dependent depend on Epsilon 0 and mu 0 and that velocity comes out to be so close to the known value of velocity of

00:10:39 light at that time that he propounded that light must be an electromagnetic wave and this was in 1865 and in 1888 hindrik Herz in Germany did experiments to generate electromagnetic waves of much lower frequency and he showed that electromagnetic waves do exist and the experiments conducted by Herz were a dramatic confirmation of the predictions of Max electromagnetic Theory and today we find electromagnetic waves all around us and we will as we will go through a little bit of discussion of these

00:11:13 electromagnetic waves we will start to understand the importance of the electromagnetic waves and our basic understanding of the physics behind the description of electromagnetic waves now before I move on to electromagnetic waves uh you have done in class 11 uh discussed about waves waves on a string for example or acoustic waves sound waves so I would like to recall some of the discussions you must have had in class 11 on waves before we move on to discussing electromagnetic waves so let's look at

00:11:47 waves waves on a string now if you take a string a long string and if you take a string and push it like pull it up like this and leave it then you will develop a wave for example you can have a wave at this time a little later what you'll find is this disturbance moves here and a little later this disturbance moves here and a little later this dist so this disturbance is moving like this and that represents the wave so what you've done is you've taken a string and pulled it up and taken it up and lowered

00:12:21 out lowered it generating a disturbance and that disturbance Moves In This Direction with a certain speed called the speed of the wave and this is a wave on a string so what is happening is the wave the wave has been generated by moving the string up and down and in that process you have given energy to the string and that energy is propagating as a wave along the string in this direction so there are two things one is the string is moving up and down with a certain velocity and energy the wave is

00:12:53 moving to the right with a certain different velocity now this is one type of wave I can generate types of waves but the most most important are the sinusoidal waves so these air waves are something like this for example this could be a function of X this is y so I have a string which is uh which is along the x-axis and and I take the end of the string and move it up and down periodically so I move the string up and down periodically and in that process generate what are called as sinusoidal waves I can describe the sinusoidal

00:13:41 Waves by the following equation the displacement of the string at any value of x and time is given by a sin KX minus Omega T now I'm sure you have discussed this in your earlier class that that the displacement of the string from the equilibrium position this is the equilibrium position of the string so the string is moved up and down periodically and in that process I generate waves on the string which are sinusoidal waves because the dependence on time and space is sign is a sign function these are sinusoidal waves and

00:14:20 a is called the amplitud of wave a is called the amplitude of the wave the maximum displacement and this quantity k KX minus Omega T is called the phase txus Omega T isort is the face of the wave and so this is let me let me explain what I what I've drawn is this shape of a string at some instant of time say t is equal to Z so this represents a sinusoid wave of amplitude a and described by a sin KX - Omega T So this gives me the displacement of the string from the equilibrium position as a function of

00:15:11 space and time so let me try to draw two pictures here one is I look at this wave at a given instant of time like here let me redraw the figure here so the general wave is described by X Y of XT is equal to a sin kx- Omega T So at T is equal to Z some arbitrary time which I call T is equal to 0 y of x t is equal 0 will be a sin KX so I what I've done is I have I take a snapshot of this string at a time which I call T equal to Z I take a snapshot and that snapshot of the string it dis placement of the string from the

00:16:00 equilibrium position is given by a sin KX so let me draw the figure again here so a sin KX will look something like this so this is the amplitude a look here the sign function varies between plus one and minus1 so it goes from plus a to minus a so this is y as a function of X it's a sinos soidal function and which repeats after every this distance this distance is referred to to as wavelength wavelength of the wave so please remember this is a snapshot of this string at T is equal to Z if I had a very fast camera I could

00:16:48 have taken a very short exposure of the string at some instant of time which I call T is equal to Z and I will see an image of the string like this this is the amplitude and this is the wavelength so you can see here the sign function at this point x was zero so the amplitude is zero the as X increases the amplitude increases like a sign function then after this distance the sign function repeats itself so in going from here to here the the quantity KX phase must have varied by 2 pi because the sign function

00:17:21 repeats itself whenever the angle changes by 2 pi so this distance must be such that KX must be equal to 2 pi so let me if I call this distance as Lambda I must have K Lambda equal to 2 pi or k isal to 2 pi by Lambda and this quantity k is called The Wave number or propagation constant so this describes it is related to wavelength through k isal 2 pi by Lambda and it defines that how much is the distance over which the period the period of wave along the distance Direction so after every distance Lambda

00:18:17 the wave repeats itself so the distance from this point to this point is Lambda the distance from this point to this point is Lambda the distance between these two points is Lambda so whenever you take two points of equal face the distance between them at a given instant of time is given by the wavelength of the wave so that's the wavelength now let me look at another picture what will the string look like from one given point so let me look at U uh at x equal to 0 so I position myself at some point

00:18:51 which I call x equal to0 and so this is at I position at this point on the string and look at this how the string varies as a function of time so y let me recall y of XT we have written as a sin KX - Omega T So at X is equal to 0 we have y of 0 t is equal minus a sin Omega T So if I position myself at a point which I call X is equal to 0 the amplitude of a string depends on time of the F of the form minus a sin Omega T So if I plot this function again this is y as a function of now time at X is equal to

00:19:42 Z goes like this this is minus a sin Omega T so this is a here and that is minus a and so as you can see here the string moves up and down and repeats itself after a certain time which I call T capital T the time period so the string goes up and down at a given instant at any point on the string and repeats itself after every time capital T So capital T must be such that the quantity Omega T changes by 2 pi in going over time capital T at this point T was Zero at this point T must be such that Omega

00:20:24 T must have become equal to 2 pi so I can write Omega * T is equal to 2 pi or Omega is equal to 2 pi by T and if I call the frequency mu is equal to 1 by T this becomes Omega is equal to 2 pi new so Omega is called angular frequency and new is called the frequency so Omega is equal to 2 pi new so this quantity Omega which appears in this equation is nothing but 2 pi * the frequency and K which appears here is nothing but 2 pi by wavelength so I can write this equation in a containing wavelength and frequency so let me write

00:21:13 Y is equal to Y of XT is equal to a sin now KX is 2 pi by Lambda x minus 2 pi Newt that is Omega T So this I can write as a a sin 2 pi uh 2 pi into X by Lambda minus so this is the equation in terms of wavelength and frequency otherwise the equation can also be written this is the same as the equation as this a sin KX minus Omega these two are the same equations this is written in terms of wave number and angular frequency this is written in terms of wavelength and frequency so please differentiate

00:22:05 between these two figures one showing the shape of the string at a given time as a function of X this is a snapshot of the string at any instant of time the shape of the string how it will look at a given instant of time and this other figure which is how the string at any point moves that means I position myself at a point on the string and see how that point on the string moves up and down as a function of time and that is as a function of time here this is a function of space coordinate here so we must be

00:22:42 very careful in analyzing these two figures one is a shape of the the displacement of the string as a function of position the other is a displacement of the string as a function of time so these two are important aspects now I want to show you that this is a propagating wave that this wave is propagating so my equation is y of XT is equal to a sin KX - Omega T so I want to draw how the string will look at a certain instant of time say t is equal to Z and a slightly later instant of time so uh let me draw how

00:23:35 the so at T is equal to z y of XT will be a sin KX which we've drawn before so this is a function of X so it looks like this so this is at t equal Z Now slightly later time say at T is equal to T1 y of XT will be a sin KX - Omega T1 so slightly later time T1 I plot I lo I look at the string y of XT is a sin KX minus Omega T1 this was at t equal 0 so I had sin KX this is a Tal T1 so I have sin KX - Omega T1 now this is the same sign function as this except that it is displaced so this was this argument was

00:24:38 0 at X is equal to 0 this argument will be zero at X is equal to Omega T1 by K so what will happen is the string will now look something like this so this is at T isal T1 please note that at X is equal 0 now it is minus a sin Omega T1 so there is a negative displacement and the function becomes zero so this point has moved to this point and this point now where is this point this point will be so suppose I call this x not X1 so this point X1 is such that kx1 minus Omega T1 is equal to Z at this point the argument of the sign

00:25:29 function was 0 KX is equal to 0 at this point kx1 minus Omega T1 is again zero the same point has moved here and so X1 and T1 are related so X1 kx1 must be equal to Omega T1 or X1 by T1 is equal to Omega by K now what is X1 by T1 what we see is the wave which was at this point this point was were here at T is equal to Z has moved to this point in a Time T1 every point on the string if you look at any point on the string it has moved by a certain distance in a Time T1 and so this must represent the velocity or

00:26:09 speed of the wave so velocity of the wave is equal to Omega by K Omega by K is the velocity of the wave so that means in this equation one is that it represents a wave a propagating wave as you can see here a little later for example if I were to draw the same figure a little later it will go like this so this is moving this side it's moving like this the entire wave is moving in the positive X Direction so at T is equal to Z T is equal to T1 T is equal to T2 and so on so as time progresses the entire wave

00:26:57 seems to be moving like this this so it represents a propagating wave and the velocity of propagation is Omega by K I can actually represent this in terms of 2 pi new omega was 2 pi new and K is 2 pi by Lambda which is equal to new Lambda so this is the equation which you must have derived velocity is equal to new Lambda velocity is related to the frequency and wavelength of the wave by V is equal to new Lambda so what we see is this particular form of an equation represents a wave a propagating wave propagating along the X

00:27:39 Direction because at T is equal to Z the wave was in certain position at a slightly later time the wave has moved in the forward direction towards positive X Direction and so this represents a propagating wave propagating along the plus X Direction now I will leave it to you to to discuss that a wave represented by y of XT is equal to a sin KX plus Omega T represents a wave going in minus X Direction so if this is a wave going in the if this is the wave going in the plus X Direction I would like you to

00:28:20 show argue like I have done that this is a wave which is going in this direction so depending on the sign between the space dependent and the time dependence part the wave is represented the wave is either going in the plus X direction or minus X Direction and that is a wave which is going in the negative X Direction so please have a argument and try to plot figures and to show that this is a wave going in the uh minus X Direction so what we have seen is this is a nice representation of a propagating

00:28:55 wave and this wave is is what is this wave after all this wave is nothing but how the string goes up and down as a function of time so there is a string which is going up and down and that disturbance on the string is actually propagating in One Direction either the plus X direction or the minus X Direction so that's a sinusoidal wave and sinusoidal waves are very important waves because uh later on in your studies you will see that any form of wave can be represented as a sum of different sinusoidal waves this is a

00:29:28 very very important Concept in physics that you can represent any wave as a superposition of different sinusoidal waves of different frequencies and this so study of sinusoidal waves is very important from the point of view of physics also note some interesting aspect here so let me draw this string again y as a function of x now if you see here the string is uh is moving like this so a little later as we draw before moved into this position so the string which was for example at this point the string which

00:30:14 was here has moved here up like this and as you can see here the maximum extension of the string is also here the the equilibrium position of the string was like this now it is stretched in this direction so maximum stretching is taking place here and the point is also moving up and down at the greatest velocity at this point so in propagating waves the potential energy of the string is contained in the extension of the string in the tension which is extending the string and the kinetic energy is

00:30:50 determined by the energy of motion of the string up and down and as you can see here the point where the kinetic energy is maximum is also the point where the potential energy is maximum so maximum stretching appears here at the point of intersection with the axis here and that is also the point where the velocity of the string is maximum in upward or downward Direction so for example you can uh you can calculate the for the velocity you can calculate Dy by DT which is equal to so let me take let

00:31:21 me draw this in in a separate slide here so let me take a wave in the plus X Direction Y is equal to Y of XT a sin kx- Omega T so if you calculate the velocity of the string is actually given by Dy by DT which is minus a Omega cos axus Omega T why the displacement of the string how does the displacement of the string vary with time and how does the stretching of the string vary with position that means how does y depend on X that is stretching and that is given by a * K * cos kx- omeg and as you can see here both of

00:32:11 them are described by the same cosine function points where the velocity is maximum where the cosine function is one it's Al the it's also the point where Dy by DX is maximum where the cosine function is one this determines the kinetic energy of the string that determines the potential energy of the string and they are in Phase so you find that in propagating waves the potential energy and kinetic energy are both in phase and as the wave propagates it carries the energy so this was a very brief

00:32:41 description of waves that you must have studied in class 11 I would urge you to go back and pick up the class 11 book and see for yourself uh different properties of waves that you must have discussed at that time you must have discussed for example waves in gas gas sound waves in a gas for example you may have discussed superposition of waves and so on so I would urge you to go back and refresh your memory as far as the waves are concerned because now we will be discussing a very very important

00:33:09 Clause of waves called electromagnetic [Applause] waves now electromagnetic waves are quite different from the waves that you must have discussed till now waves on a string for example you must have seen waves on a on a water surface so in a string what happens is the string is moving up and down and in that process there is a wave which is moving so the string is not moving in the forward Direction string is only moving up and down and the wave is going in a certain direction now remember these are called

00:33:53 transverse waves because the string moves up and down in the vertical Direction while the wave moves in the horizontal Direction the direction of propagation of the wave is perpendicular to the direction of motion of the string and so this is the transverse wave so I could have the string moving up and down and the wave propagating like this or I could have a string going forward and backward and the wave propagating like this so these are two different types of transverse waves one which is displaced along the y direction

00:34:19 and propagating along X the other is displacement along Z Direction and propagating along X so you can have two different transverse waves sound wave represent what are called as longitudinal waves where there are compressions and RAR factions which are propagating in the wave so when I speak I am generating sound waves so there are waves of compressions and ractions in the air which is surrounding me here and those compressions and ractions are such that air molecules are so for example when I speak when I speak

00:34:47 the air molecules are moving like this and the sound wave is propagating the forward Direction so these are called longitudinal waves the displacement of the particle is along the direction of motion of the wave now electromagnetic waves are also transverse waves now unlike in a waves on a string which requires a string for propagation of the wave or sound waves where you need gas or some kind of a medium in which you have to propagate electromagnetic waves can propagate in free space light is a form of electromagnetic wave

00:35:42 and we are getting light from the sun we are getting light from Stars which are millions of light years away so we're getting light from all around the place and please remember there's hardly any anything in between the uh outside the solar system and the solar system between the the stars and us or between Sun and us so these waves are able to propagate in free space and so they are a completely different type of wave they don't need a medium to propagate of course the presence of a medium can

00:36:08 change their propagation properties but you don't need a medium to propagate and so these electromagnetic waves are characterized by Electric variation of electric and magnetic fields in space so these are the waves which are electromagnetic waves are waves in electric and magnetic fields they are represented by electric and magnetic fields and not displacement of a string or pressure in a gas they don't need a medium and they are actually nothing but electric and magnetic fields which are

00:36:56 propagating in space carrying energy and momentum so before we discuss more details what I would like to do is to draw a figure to show you what how electromagnetic waves are represented and describe a bit about the figure so that you understand very clearly what is the meaning of the figure that we plot okay so couple of things which I need to mention before I plot the figure the electric and magnetic fields are perpendicular to the propagation direction that is why they are called transverse waves the electric and magne

00:37:58 fields of the wave are perpendicular to the propagation Direction electric field is perpendicular to the magnetic field and this Vector e cross B is along the direction of travel of the B so electric and magnetic fields are perpendicular to the propagation Direction electric and magnetic fields are perpendicular to the propagation Direction electric field is perpendicular to the magnetic field and electric field cross magnetic field is along the propagation Direction so that means electric field magnetic field and

00:39:11 propagation Direction form a right-handed coordinate system and also electric field and magnetic field are in Phase so the electric field and the magnetic field are always in phase in these propagating waves just like I showed you in the string the kinetic energy and the potential energy are in phase in the propagating wave here the electric and magnetic fields are in Phase so let me draw a figure which represents an electromagnetic wave and then try to describe to you what this what is the meaning of that

00:40:03 okay so let me draw the uh the electric field so this is the electric field and let me draw the magnetic field which will look like this so let me draw some vectors here so these are they I represent electric Fields magnetic fields are perpendicular so please note that so if I call this uh x y z so this is the electric field and this is the magnetic field and that is so please note this figure what I've drawn is how does the electric field vary with position this is at some instant of time remember just like I

00:41:19 drew for the waves on a string I take some instant of time which I call T is equal to0 and plot how they electric field varies with position and how the magnetic field varies with position so you can see here first thing is the electric field the wave is actually propagating along the Z direction we propagating along Z direction we will discuss that and the electric field is perpendicular to the propagation Direction the magnetic field is perpendicular to the propagation Direction so the wave is a transverse

00:41:52 electromagnetic wave it's a transverse wave because electric fields and magnetic fields are perpendicular to the propagation Direction second the electric field and magnetic field are perpendicular to each other so electric field is perpendicular to magnetic field at every Point here here everywhere electric field and magnetic field are perpendicular to each other and electric field so electric field is pointing along X Direction in this figure magnetic field is pointing along y direction so e cross B which is X cap

00:42:28 I cap cross JC has to be KC and so the wave is propagating along that direction so electric field magnetic field and the propagation Direction form a right-handed coordinate system XY Z electric field magnetic field propagation Direction okay now one has to be a little careful in rep in sort of trying to figure out this figure because it's a very abstract figure in the following please note that these arrows only represent the magnitude and direction of the electric and magnetic fields at every point on the axis of this

00:43:13 wave so this Arrow implies that the electric field at this point has a large magnitude and is and is pointing in the upward Direction at this point the electric field at this magnitude and pointing in downward Direction at this point the magnetic field is along the y direction and has this magnitude so all these fields are actually fields at various points along the axis and these are the magnitude and direction of the electric and magnetic fields please remember that these arrows only represent the magnitude and

00:43:49 direction of the fields and do not represent any displacement of any object in the case of the vibration of a string waves on a string the the the Y versus X was actually a plot of displacement of the string as a function of position at different points so the the position of the string was being marked there here in this figure there is no displacement there is nothing connecting these two points this Arrow represent the fact that at this point the electric field has this magnitude and direction

00:44:26 at the same point the magnetic field has so much magnitude and Direction similarly here the electric field has the so much magnitude and direction this magnetic field has so much magnitude and Direction so these are all nothing but electric and magnetic fields represented along the axis but by these arrows and they do not this line does not represent a displacement of a particle or a displacement of a medium or anything so this one must be very very careful and this whenever you look at at a figure of electromagnetic waves please

00:44:59 keep this in mind that unlike the case of vibrations of a string there is nothing which is displacing up and down all that it implies is as the wave propagates for example like this at this point if the electric field is if the wave is propagating like this if the electric field is pointing up magnetic field is pointing here and at this point if the electric field has a certain magnitude there is a certain magnitude of magnetic field at a little later time maybe the electric field is pointing down and the magnetic field is pointing

00:45:27 another Direction so you have essentially the electric and magnetic fields at this point will vary with time so for example electric field will start from zero increase in magnitude in the upward Direction and then it can become zero and increase as a negative Direction at this point and oscillate with time for a sinusoidal electromagnetic wave which I will write down in the equation at every point the electric field and magnetic fields will vary sinusoidally with time so if I look at a point if I position

00:46:00 myself at a point and if I had a detector for electric field that detector will tell me that as a function of time the electric field is varying it starts from some value keeps on increasing to a maximum then becomes zero then starts to increase in the negative Direction becomes a maximum becomes zero again and periodically oscillates like simple harmonic motion but there is nothing moving it is simply the magnitude and direction of electric Fields similarly at the same point when the electric field is increasing

00:46:29 magnetic field is also increasing but in the perpendicular Direction so the electric field is pointing up the magnetic field is pointing towards me so that e cross B is in the direction of motion if the electric field is pointing up and magnetic field is pointing towards you the direction of propagation must be here so please remember electric and magnetic fields are perpendicular to propagation Direction electric and magnetic fields are both perpendicular to each other Electric field cross magnetic field vectors must be along the

00:47:00 propagation Direction and this figure is a very important figure that you will find in your textbook and at every place this is a figure which represents this a bit of abstract figure this line does not represent any motion of or displacement of any object all it says is it's a point connecting the tips of the electric Vector as a function of position here and the tips of the magnetic vectors are the functional position so at this point when the electric field has this magnitude magnetic field has this magnitude at

00:47:29 this point if the electric field has this magnitude magnetic field has this magnitude so keep this in mind when you are looking at this figure like this and this is a very very important aspect that we must remember when we are looking at such figures now let me write down an equation now I in principle we could use those maximal equations and derive an equation which predicts the existence of these waves but that is beyond the scope of this course here so what I will do is I will write down the solution

00:47:58 of the those equations in the form of an electromagnetic wave and show you that those equations are consistent with the max equations that we have written earlier so I will write the solutions and show you that those Solutions are representing waves which we call as electromagnetic waves and those Solutions are consistent with the max equations we wrote earlier so let me write down uh electromagnetic so sinus oid waves and these are electromagnetic waves again so um if I so let me draw the figure again

00:48:48 here so I had x y and z so I drawn the electric field wave like this and magnetic field waves magnetic field is in this plane the electric field is in this plane so I will write for example e is equal to I cap e0 sin k z minus Omega and V Vector is equal to JC B sin so these are equations which are very similar to the equations we had written for waves on a string we have to write for electromagnetic waves we have to write an equation for the electri field as a function of position and time and the

00:50:00 magnetic field as a function of position and time this is a wave propagating along the Zed Direction please remember in the case of the waves on a string I had written it as KX minus Omega T which represented waves along the X Direction here I'm writing k z minus Omega T that mean this is a wave propagating in this direction in the plus Z Direction these are sinusoidal waves because they are sign functions as you can see both are in Phase because both are sin KZ minus Omega T their directions are such that if e is

00:50:39 along i b is along jcap so e and B are perpendicular to each other and both of them are perpendicular to the propagation Direction which is along zc Direction wave is propagating along Zed electric field is along so electric field is along this direction this is the electric field Direction and this is the magnetic field Direction here so magnetic field seems to be in the magnetic field is in the YZ plane the electric field is in the XZ plane they are in phas they are perpendicular to each other and E cross

00:51:18 B is nothing but I cross J which is along Kap Direction which is the propagation Direction so as before K is equal to 2 pi by Lambda Lambda is called the wavelength of the electromagnetic wave and Omega is equal to 2 pi mu new is called the frequency of the electromagnetic wave so in this K is 2 pi by Lambda Lambda is the wavelength of the electromagnetic wave Omega is 2 pi new new is the frequency of the electromagnetic wave K is called the propagation constant or wave number and Omega is called the angular frequency of

00:52:23 the electromagnetic wave so the electromagnetic waves are now described by these and these are one type of solutions of electromagnetic waves these are called sinusoidal waves and uh these are this represents one type of wave now here as we have discussed before this quantity the velocity of the wave is equal to Omega by K this ratio Omega by K represents the velocity of the wave and these two represent sinusoidal waves electromagnetic waves the first equation represents the electric field variation

00:53:07 with position and the second equation represents magnetic field variation with position now I want to uh before we discuss these waves in more General I would like you to show a figure a slide which shows you the type of electromagnetic waves at various frequencies and various wavelengths so this is the electromagnetic spectrum this is the Spectrum here uh in this direction the frequencies are increasing which means the decreasing wavelength here as you can see frequency and wavelength are inversely related to each

00:53:37 other the velocity is a product of frequency and wavelength so as the frequency increases from left to right in this figure the wavelength decreases so you can have waves of different uh frequencies and different wavelengths U here so there are starting from very low low frequencies electromagnetic waves can occupy where frequencies of very high values so there have we have given different names for waves of different frequencies here you have what are called as radio waves with frequencies of the order of a megahertz 10 6 Herz

00:54:12 then you have microwaves with frequencies of the order of a gahz 10^ 9 Hertz then you have an infrared region here which is which has wavelength which has frequency slightly less than visible and this is the visible region of the electromagnetic spe spum so this is the complete electromagnetic spectrum the visible wavelengths are here and the frequencies range from about 4 10 14 Herz to 7.5 10 14 Hertz and then comes ultraviolet in this region about 10^ 16 Herz frequencies then we have xrays which are 10 per 18 Herz frequency and

00:54:46 then we have gamma rays which are 10 per 20 HZ frequencies so you can have electromagnetic waves of varying frequencies and the visible spectrum to which our eyes are sensitive forms a very small fraction of the entire electromagnetic spectrum so this is as the frequency increases here the wavelength drops down and you can actually calculate and show the wavelength of these waves are much much shorter than the wavelength of these waves and I will leave leave it to you to calculate the corresponding

00:55:14 wavelengths knowing the speed of these waves because we will now in the next class what we will do is discuss in more detail starting from this equation which I wrote down of the electromagnetic waves elect and magnetic fields I will show you that these two equations are consistent with the Max's equations and I will show you that Omega by K which is the velocity of these waves is related to Epsilon 0 and mu 0 the dialectric pertivity of free space and dialectric permeability of free space Epsilon 0 and mu 0 and that is where

00:55:48 Maxwell connected electricity and magnetism and Optics he showed that Optical waves must be electromagnetic waves because he found out that the speed of these waves are is so close to the speed of light waves that light must be electromagnetic wave so I will stop my class here right now and we will continue with our discussions starting from these two equations and I will show you we will calculate the relation between electric and magnetic fields here and I will show you that the velocity of the waves of

00:56:17 these waves is nothing but the velocity of of light in free space thank you very much [Music] n [Music]