 Hi everyone, welcome back. In my last lecture, we discussed blackbody radiation spectrum. We talked about how we can construct a blackbody using some kind of a cavity shaped structure. And then we looked at the experimental observations associated with the blackbody spectrum arising from such kind of a cavity shaped structure. Namely, we talked about this particular graph. If we study the energy density coming out of some sort of a cavity shaped structure or more specifically the thermal radiation emitted by objects, then that kind of a spectral energy density or the energy emitted per unit time, per unit surface area with respect to frequency has this kind of a variation where the energy density is very low for low frequencies, increases sharply for increasing frequencies, reaches a maximum and then again goes to 0 as frequency tends to infinity. Now the thing is, this is an experimental observation based on experiments done by scientists over many decades. Now the question is how can we explain this kind of an observation? You see the whole idea behind physics or behind a scientific theory is to explain why things are the way they are. So the natural question is can we use the principles of physics, the laws of physical theory to explain why the blackbody spectrum or the thermal spectrum coming out of objects is of this particular shape. One of the earliest attempts was made by Rayleigh and Jeans where they used classical theories of physics to try to explain this kind of a distribution. And in today's video, that is what I am going to concentrate in. In today's video, I am going to derive the Rayleigh, Jeans law using the classical approach and I am going to try to obtain some sort of an energy distribution expression that will give us a variation with respect to frequency and then we will make a comparison with the experimental data and we will see something very, very interesting. We will see that there is a serious conflict between the classical explanation of blackbody radiation and the experimental observation which is what is going to lead to our next video which I am going to publish after couple of days regarding the Planck's postulate or the Planck's explanation of blackbody radiation spectrum. But before we go there, first in today's video, I want to discuss the Rayleigh, Jeans law and how we can calculate the energy density associated with blackbody radiation with respect to frequency. Now the derivation itself is slightly lengthy. Therefore, I have divided the derivation into three different approaches which is basically based on what Rayleigh, Jeans did. So initially, they tried to use classical electromagnetic theory to explain why standing wave patterns can form inside a cavity. Then they used some kind of a geometrical arguments to calculate the number of standing waves that can be formed inside a cavity for a given frequency range and then they used equipartition theory of energy to calculate the average amount of energy associated with a standing wave. So if you can calculate the total number of standing waves formed inside the cavity for a given frequency interval, multiply it with the average amount of energy associated with a standing wave. Then in principle, you can calculate the energy density associated with a given frequency range with respect to frequency. So these are the three main steps that we also are going to follow today. So let's elaborate on them one by one. So first of all, based on classical electromagnetic theory, we can understand why some kind of a cavity will emit thermal radiation or electromagnetic radiation in the first place. So here we have some kind of a cuboid shaped metallic cavity whose dimensions are a, a and a along x, y and z axis respectively. It is hollow from the inside and the inside surface is metallic and it emits some kind of a electromagnetic radiation which is what the blackbody radiation spectrum is coming from. Now the question is why are electromagnetic radiation being emitted from the walls? Well quite simply put, if this kind of a metallic cube has some kind of a temperature, then because of the temperature, the molecules, the atoms on the walls of the cube, they will have electrons and charged particles which are going to be agitated due to this kind of a heat contained within the body and because of this thermal agitation, they are going to oscillate or vibrate and then they are going to emit these kind of electromagnetic oscillations which are what is going to constitute the radiation coming out of the cavity. So it is not really difficult to imagine that most charged particles on the walls of the cavity are going to emit thermal radiation or electromagnetic radiation because of thermal agitation due to the heat contained in the body. Now if we look at the thermal radiation that is being emitted by all the walls inside the cavity, then we can look at three distinct directions along the x-axis, y-axis and z-axis because there are walls, metallic walls along each of these directions, all these radiations emitted along x, y and z-axis will be oscillating back and forth between parallel walls along x, y and z-axis. Now what is an electromagnetic radiation? It is essentially electric and magnetic oscillations. So if we look at some kind of an electromagnetic radiation going from one end of the cube to the other end, then the electric field associated with the electromagnetic oscillation is perpendicular to the direction of propagation. That means the electric field that is associated with some oscillating electromagnetic radiation is parallel to one of the walls. Because of this particular reason if you look at the electric field at the very wall themselves the electric field actually turns out to be zero. This is because according to electromagnetic theory whenever you have some kind of a metallic surface if there is a non-zero electric field on the metallic surface then the charges in the metallic surface will rearrange themselves so as to compensate for that and cancel out the electric field as a result of which the electric field on a metallic surface comes out to be zero. So if the electric field on the metallic surface comes out to be zero that means these electromagnetic oscillations will have nodal points on the walls themselves. Let us try to understand this by first studying a one-dimensional cavity and then elaborating on further to three-dimensional cavity. So let me first draw a one-dimensional cavity having walls only along a particular axis. So let's suppose that I have here a one-dimensional cavity where the first wall is at x is equal to zero and the second the next wall is at x is equal to a and as I just now said the electric field on the wall itself has to be zero because of this reason we end up getting standing wave patterns that has nodal points on the walls at both the ends. So for example we can have a wave that looks like something like this where the distance between the walls will be equal to half the wavelength of that standing wave. Another wave that we can have is something like where the distance between the walls will be equal to exactly one wavelength another one that we can have will be something like something like this. So you can see that we can associate some kind of an integer with all possible standing waves that are possible. So for example let's suppose this corresponds to n is equal to 1, this corresponds to n is equal to 2, this corresponds to n is equal to 3. I can come up with a condition so for example since we are discussing a one-dimensional cavity right now the condition for standing waves is quite simply put the dimension of this cavity which is a is equal to n times lambda upon 2. Where lambda is the wavelength of the standing wave n is an integer and can take values of one two three four five up to infinity. There are infinite possible standing waves but only those which corresponds to this particular condition. Further I can also write the whole thing in terms of frequency. So because lambda is equal to c upon nu so I can say n lambda. So lambda can be written as c upon nu upon 2 nu or I can also write down the expression in terms of n itself. So n is equal to 2 a upon c nu alright. So here a is the dimension of the cavity n is the integer nu is the frequency lambda is a wavelength. Now what is our objective? As I already told you our objective is to calculate the total number of standing waves that is possible for a given frequency interval right. This simply gives us an idea about all the standing wave patterns that are possible for different wavelengths or different frequencies but I am interested in for a given frequency interval because I want to plot the energy density for some given frequency interval. For that what we are going to do is we are going to draw an axis for n. n is associated with these numbers right these integers we are going to draw an axis and there we are going to look at the number of points between some frequency interval. So the axis would look something like let's suppose this is an axis and the origin is n is equal to 0 which obviously does not associate with any kind of a wave but then we have n is equal to 1 then we have n is equal to 2 then we have n is equal to 3 n is equal to 4 5 6 and on and on. So let's suppose in this line number line or axis you have these points and every single point corresponds to some allowed value of n and for every allowed value of n it corresponds to some allowed standing wave that is possible inside the one-dimensional cavity. So for a given frequency range what do I mean by frequency range? So let's suppose I take a frequency nu which is some random frequency and then I take a small increment nu plus d nu. For a given frequency range what are the number of standing waves that are possible? For that all we are going to do is let's suppose that there is some kind of an integer corresponding to nu alright so if there is some kind of an integer corresponding to nu in this kind of a number line let's suppose this is the one. Let's suppose I call this simply as n okay n is equal to how much? 2a upon c nu. What if I take a small increment in the frequency in that case I can go slightly forward so this is the range of n for some given frequency and for some further value if I increase the frequency we will also get a bigger range for n. Let's suppose I get n plus dn or something like that which will correspond to 2a upon c nu plus d nu alright. So there are going to be certain number of points in this particular range the moment I go from frequency to frequency plus d nu and those number of points will be associated with some possible standing waves. Now what is the number of those possible points or possible standing waves? Well the number of the possible points allowed integers or possible standing waves in a given frequency interval clearly is the difference between this and this right. So this minus this will give us the total number of points because essentially we are looking at it in the n axis and the distribution of the points is kind of uniform. So the difference between this and this is basically going to give us 2a upon c times d nu and it is kind of obvious that the number of points is not dependent on frequency but dependent on the frequency interval because the greater the frequency interval greater would be the number of points but it has no relationship whatsoever on the actual value of frequency because whether the frequency is low or high it doesn't really matter because these points are uniformly distributed. So it is the interval that is affecting the total number of points or total number of standing waves that I'll have in this kind of a one-dimensional cavity. So if I have this many number of integers which are possible or standing wave patterns what is the actual number of waves the actual number of waves will actually be twice this why because if you look at each individual wave every single wave in the cavity is associated to two polarization why so if you look at this kind of a let's suppose along the x-axis two different walls you will have oscillation that is going like this whether electric field is along the z-axis you have another oscillation which is going like this whether electric field is along the y-axis. So there are two perpendicular directions perpendicular to x-axis z and y so the electric field can be either along z or y-axis so essentially there are two distinct polarizations possible compared to every single standing wave and those two polarizations will be associated with two independent waves so essentially I will multiply this to get the actual number by 2. So finally for every single frequency interval we end up getting 4 a upon c d nu. So this will give you an idea about how we are approaching the problem we are trying to look at some kind of a cavity and trying to find out the condition for these kind of standing wave possibilities and then by drawing an axis for these integers the allowed numbers n we are trying to look at an interval and trying to find out the number of these points possible in that interval multiplied by 2 to get the actual number of waves. We are going to follow a similar approach for a three-dimensional cavity where in a three-dimensional cavity we will have three distinct directions so the waves can exist in three distinct directions we can have standing waves in the x-axis standing waves in the y-axis standing waves in the z-axis and we will have three different numbers nx ny nz for all these three distinct directions and using those condition we will try to obtain the same thing the number of standing waves possible in a frequency range. So let me first rub the board. Alright so for 3d cavity calculations we are going to have three distinct directions x y and z-axis and every single direction will be associated with some kind of a standing wave condition. So along the x-axis we will have a similar kind of a condition where the length of the wall or length of the cube along the x-axis because it's a cube all the walls have the same dimension so a will be equal to some integer I'm going to call this as nx times lambda x upon 2 so lambda x upon 2 is the wavelength of the electromagnetic radiation along that particular direction. Now I can associate this with something more I'm going to call this as kx which is the wave number so you must be familiar with what wave number is so wave number is actually k which is equal to 2 pi upon lambda so for the sake of ease of calculations I'm going to use this particular number so associated with it with lambda x I'm going to have 2 pi upon kx so we are essentially getting nx pi upon kx which can be rewritten as nx is equal to kx a upon pi or a upon pi kx where what is nx nx is essentially integer that can take the values of 1 2 3 up to infinity and kx is a wave number associated with the possible standing wave similarly I can have a similar calculation for y-axis so for y-axis we will have again the same dimension and a is equal to ny lambda y upon 2 which is equal to ny pi upon k y hence we will have ny is equal to a upon pi k y where ny is equal to 1 2 3 and on and on all the integers and lastly we will have the z-axis along which we will have again a similar condition a is equal to nz lambda z upon 2 which is equal to nz pi upon kz or we can say that nz is equal to a upon pi kz where nz is equal to 1 2 3 and all the integers now if I combine all these three distinct equations I should get something like this so if I write an equation for example nx square plus ny square plus nz square so this will be equal to a upon pi square kx square plus ky square plus kz square which is essentially equal to nx square plus ny square plus nz square which is equal to a upon pi square kx square plus ky square plus kz square is k square so kx ky kz is associated with the wavelength components along x y z-axis and k is basically the overall wave number corresponding to some kind of a wavelength that can be constructed with respect to the x y z components so this should be k square all right so this I can write as nx square plus ny square plus nz square is equal to so if I take the square out from the equation I will write this as root over and this becomes a upon pi but what is k? k is essentially equal to 2 pi upon lambda so here pi and pi gets cancelled and we are left with 2 a upon lambda and this can further be simplified in terms of frequency as I can write the frequency as so this is equal to 2 a upon c upon nu lambda is c upon nu so therefore finally I end up getting this particular expression that 2 a upon c nu is equal to root over nx square plus ny square plus nz square so this is the condition associated with the standing waves that are formed in a three-dimensional cavity now I have to make something clear actually the detailed derivation requires that we choose a standing wave that is not necessarily along the x y z direction but in a random direction and its components are along the x y and z axis and then we apply the conditions for the components and finally come up with this expression if you write them in terms of components you will have a certain angles associated with the x y z axis and there will be a component angle associated every single term I did not write it in that fashion because ultimately even if you do the calculation from that perspective you come up with the same condition all right a more general sort of a derivation would include a random direction along which the wave is propagating and its components along x y z axis so that there is some kind of a cos alpha beta gamma component associated with lambda x lambda and when you combine all of them you come up with the same condition I tried to simplify it because anyways it was getting a little bit of a lengthy derivation so I just simplified it because ultimately anyways I am getting the same condition that I would have gotten from that perspective also so this is the condition that is associated with all the standing waves that are formed in a three-dimensional cavity now what can we make from this you remember my previous calculation for one-dimensional cavity we drew an axis for the n and then we tried to figure out the number of points in a frequency range so we are again going to do the same thing here I am going to draw three axes corresponding to nx ny and z and then I am going to try to make a calculation of finding out the number of points in a frequency range of d new so let me make the diagram first all right so this is the best diagram I could make with my hands so essentially we have three axes let's suppose that this is nx all right let's suppose this is ny and this is nz so essentially three axes correspond to nx ny and nz and because nx ny and z can take all integer values so we only have these integer values uniformly distributed along these three different axes now if we look at this construction what this construction essentially leads to is all possible combinations of nx ny and z as long as they are positive integers so all possible combinations of nx ny and z which are positive integers essentially creates this three-dimensional grid all right there's a three-dimensional grid in the first quadrant let's suppose where every single integer values corresponds to some possible standing wave in the three-dimensional cavity now if we want to figure out what are the number of these points for a frequency interval first we must analyze this expression if you analyze this expression where nx ny and z are these three distinct perpendicular axes and now what does this look like this looks like the surface of a sphere so for example the surface of a sphere is what x square plus y square plus z square is equal to r square where r square is the radius of a sphere now if we look at this particular expression in the right-hand side so this particular expression in the right-hand side think of this as r okay think of this as r corresponding to some sphere or spherical surface associated with nx ny and z so nx ny and z can change in such a manner that along the surface of a sphere the value of the frequency will remain constant so along the surface of a sphere the values of nx ny and z may change but the value of frequency will remain same so for every distinct frequency we can have multiple values of nx ny and z but those will all lie on the surface of a sphere so for example i have this one spherical surface i don't know if it looks like a sphere to you or not just imagine that along these axes nx ny and z we have some kind of a grid containing uniformly distributed points corresponding to nx ny and z but because of this particular expression i am drawing some kind of a spherical surface so let's suppose this particular spherical surface the inner spherical surface this corresponds to some frequency new so for this particular spherical surface it corresponds to let's suppose 2a upon c times new because along this particular surface we can tweak the values of nx ny and z which are lying on the surface because they will all end up satisfying this particular condition for this particular fixed frequency now if i increase the frequency what is going to happen if i increase the frequency this spherical surface will only increase in size right this will just increase in size so if i take a small increment in frequency the spherical surface will slightly increase in size that's all so i have drawn another surface so this surface let's suppose is associated with 2a upon c new plus d new all right so for frequency new we have this surface the inner surface and for new plus d new we have the outer surface now what are we interested in we want to calculate all these points that can exist between these two surfaces that means this is a shell right this is a shell and inside that shell what is a number of points that is possible because they will correspond to possible standing wave patterns inside the three-dimensional cavity to do that calculation we will have to calculate the volume of this shell right so how can we calculate the volume of this shell so based on our assumptions what do we have we have that r is equal to what 2a upon c new right if r is equal to 2a upon c new then what is the value of let's suppose dr dr is equal to 2a upon c d new right now if i want to calculate the volume of this kind of a shell which defines some kind of a frequency interval then i am interested in figuring out the volume let's suppose dv dv is the volume associated with this spherical shell all right what is the volume of a spherical shell well the volume of this spherical shell is 4 pi r square so 4 pi r square corresponds to the surface area times dr so dr we already know what dr is r also we know if i substitute these values here then i should end up getting 4 pi r square is equal to 2a upon c new square and dr is equal to 2a upon c d new which comes out to be 32 pi a upon c cube new square d new now this is the volume in the shell all right now as i just now told you nx ny nz can only take positive integer values right but the sphere can be like a three-dimensional sphere that may also include negative integer values i don't want the negative integer values i don't want nx is equal to minus 1 minus 2 minus 3 right because that will not serve our purpose for this particular problem so i want to exclude all of that so i only want to take the first quadrant so imagine the three-dimensional space is divided into 1 2 3 4 5 6 7 8 quadrants right so i only want to take the first quadrant and the volume of the shell in the first quadrant because that will correspond to positive values of nx ny nz so in that situation for the first quadrant the volume of the shell comes out to be this divided by how much 1 upon 8 which essentially comes out to be 4 pi a upon c cube new square d new so now what is finally the number of standing wave patterns in a given frequency interval finally the number of standing wave patterns in a given frequency interval is equal to the number of these points corresponding to integers of nx ny nz in this shell volume so the number of points corresponding to positive integers of nx ny nz in this shell volume is equal to the volume of the shell in the first quadrant times the density of these points now the density of these points is what one because they are all going in increments of one so the density is equal to one so the density times the volume density is equal to one so this is essentially equal to exactly this value 4 pi a upon c cube d new now again as i already said in my last derivation also that every single standing wave pattern is associated with two distinct polarization right because if you have a wall and you have a standing wave then standing wave electric field can go either in this particular axis or in this particular axis so there are two perpendicular axes along which the electric field vector can change therefore there are two distinct polarizations associated with every standing wave condition possible so therefore we have to multiply this with two to actually get the number of standing waves so therefore the number of standing waves i forgot to write new square so there is a new square here which is very important by the way because that is the quantity which is going to bring all the difference in this calculation so 4 pi a upon c cube new square d new times 2 because of two distinct polarization so this is actually going to be equal to 8 so there you have it for a three-dimensional cavity the number of possible electromagnetic waves for a given frequency range is equal to this particular expression which gives us an idea about how many standing waves are there in a frequency range so now let us move ahead to the next step in our calculation which is figuring out the average energy corresponding to standing waves all right so the next step in the calculation of the black body radiation energy density using classical physics is to calculate the average energy associated with each standing wave see if we know the number of standing waves in a given frequency range we don't really need to calculate the energy associated with every single standing wave right we just need to know the average energy associated with all standing waves if we know the average energy associated with all standing waves and the number of standing waves possible we multiply them to get the total amount of energy density right so for that Rayleigh and jeans used the classical thermodynamics or classical kinetic theory where we know the law of equipartition theorem which gives us an idea about the average amount of energy associated with a harmonic oscillator you see the standing waves are generated by what they are generated by charge particles the cavity has these metallic walls these metallic walls has electrons because of heat these electrons are experiencing thermal agitation and because they are experiencing thermal agitation they are vibrating or they are oscillating they are behaving like harmonic oscillators and because they are behaving like harmonic oscillators they emit electromagnetic oscillations so amount of energy that is of the standing wave is coming out from the harmonic oscillator which is essentially the electron so these harmonic oscillators they have some sort of an oscillation and some kind of an energy associated with it and that energy is released in the form of the standing wave so if I can find out the average energy of these harmonic oscillators I can find out the average energy of these standing wave patterns because they are ultimately coming from these harmonic oscillators these oscillating electrons so equipartition theorem of energy tells us what it tells us that for a harmonic oscillator the average energy that is associated with each degree of freedom is equal to something called half k t you must have seen this expression before when you study maybe thermodynamics that k here is a Boltzmann constant t is the temperature and the average amount of energy associated with each degree of freedom is half k t but the harmonic oscillator is a unique case because the harmonic oscillator has two different energy contributions one coming from the kinetic energy and one coming from the potential energy the kinetic energy energy contribution is a result of its momentum and the potential energy energy contribution is a result of its position right kinetic energy is a result of its momentum and the potential energy is associated with its position so both of these two components are associated with half k t so therefore for a harmonic oscillator for the unique case of a harmonic oscillator the average energy comes out to be half kT plus half kT is equal to kT. This is what classical kinetic theory tells us that for a harmonic oscillator the average energy is this much. Therefore, the standing waves also should have an average energy this much. Therefore, because a total number of standing waves in a given frequency range is this much. So, what is the energy density? Finally, we are ready to calculate the energy density that we have plotted here. So, let's see what kind of result we get. What is energy density? Well, energy density is essentially the number of standing wave pattern multiplied by the average energy divided by volume per unit time. Initially, when I calculated the black body radiation, we did it for a cavity hole. So, we said per unit surface area, but because the black body radiation coming out of the hole is essentially coming out of the cavity. So, the black body radiation coming out of the hole is essentially sampling of the radiation inside the cavity. So, the average energy for the cavity will be per unit volume. So, what is a cube? A cube is nothing but the volume of the cubical cavity. So, if I divide the entire thing, let's suppose by volume, then this will get cancelled and ultimately we will end up getting 8 pi and then we will have nu square upon c cube d nu, right. So, this is the n nu d nu upon volume multiplied by average energy, all right. KT, this will give us, ok. This is the number of standing wave patterns per unit volume multiplied by the average energy which will give us the energy density for the black body radiation inside the cavity. The energy density which is per unit volume per unit time is the total amount of energy which is obtained by multiplying the number of standing wave patterns in a frequency range with the average energy which finally comes out to be this particular expression 8 pi KT upon c cube nu square. There you have it. Finally, we have obtained what is the energy density for a frequency interval d nu this much. Now, in some of the books, they will have probably written this expression in terms of wavelength. You can easily do that because nu is equal to c upon lambda. So, you can write the whole expression in terms of wavelength also, no issues, but the results are going to be the same. So, finally, we have this particular expression which is what we were interested in. Finally, classical mechanics or classical physics by using classical electromagnetic theory, by using all these geometrical arguments, by using the understanding of classical thermodynamics has come up with a derivation to explain the blackbody radiation spectrum. Let us see if it is able to explain this radiation spectrum or not. 8 pi K, these are all constants. C is a constant, T is a constant, d nu simply represents the frequency interval. So, clearly the energy density is related to directly proportional to nu square. The energy density of blackbody radiation spectrum in terms of frequency is dependent upon nu square. What is nu square? If I plot it, how is it going to look? Well, this is how it's going to look. There you have it, a parabola. This is the classical prediction of the blackbody radiation spectrum or we can say the Rayleigh genes law or distribution or the classical explanation of blackbody radiation spectrum while this one is the experimental observation. Does the experimental observation match the theoretical prediction? Does it? It matches maybe for low frequencies. For low frequencies, the theoretical predictions of classical physics matches with the experimental results, but with increasing frequencies immediately diverges and goes to infinity. It can never really go to infinity. It's not really practically possible because the energy emitted by any kind of an object is finite. First of all, so the classical physics can never really explain why is it going to infinity and for higher frequencies again the whole thing should go to zero in the first place. So, this is known as the ultraviolet catastrophe. Ultraviolet catastrophe because it is a catastrophe in physics that some of the most powerful branches of classical physics has been used to calculate an expression to explain blackbody radiation spectrum, but it failed in such a spectacular manner. And well, this is what I was talking about. This is where when all theories fail, new theories must come to explain these things. This grossly unrealistic prediction of classical theory, which is the Rayleigh-Jeans law, the distribution, as opposed to the actual experimental observations, will demonstrate that there is something deeply missing in our understanding of physics. We will see in our next lecture where Max Planck would come and provide a resolution to this particular catastrophe, the ultraviolet catastrophe. But to make that resolution, he will have to make certain assumptions and those assumptions are going to play a fundamental role in the birth of quantum mechanics. So, I'll see you in the next video where we will talk about the Planck's postulate and how Max Planck resolved this particular problem making certain kinds of very necessary and very important assumptions. So, that is all for today. I'll see you in the next video. Thank you very much. Bye-bye.