 Let us get to the black body radiation now we will not spend so much time but we will discuss all the questions whatever we get. So now that we have realized that the radiation energy emitted is a function of it is going to be a function of we have not taken up these issues it is going to be a function of temperature we had fixed the temperature if I vary the temperature it is going to vary material of the body whether it is a conductor or non conductor condition of it is a surface that is whether it is a polished or rough surface okay. So these are all the factors or the parameters which also affect the radiation of the amount of the radiation energy okay. Now if I take a given the amount of the radiation energy emitted no it is it is it is nothing copy paste problem so this has to be removed see we tend to copy paste while making this notes no I have missed this out this is the same sentence as this okay so because fonts and all I can manage so they while typing that has come out like that okay. Now as I keep saying ideal cases are always our references what is the what is a body which gives the maximum MSU power that is what we say it has that is the standard and that is the black body and that was what was black body what is a black body first let us define the black body. Black body absorbs all the incident radiation regardless of it has no preference for wavelength and direction so far we were harping on wavelength and direction but for black body it has no preference it is independent of wavelength and direction and for a prescribed temperature and wavelength no surface can emit more energy than the black body okay and although the radiation emitted by a black body is a function of wavelength and temperature stated same thing again it is independent of direction what does this mean it is also a what it is also a good emitter it is the best emitter and it is a best absorber okay it is set diffuse because it is independent of the word diffuse is used here because it is independent of direction is that okay so of course this is just to show that black body is independent of direction this is having a preference with direction that is real body okay so now coming back to Stefan Boltzmann law this has been given to us just like that so we are also stating that Stefan Boltzmann law says that E b equal to sigma t to the power of 4 we have done this already this relation this was again theoretically verified it was first given by Stefan and theoretically verified by Boltzmann and this gives the maximum MSU power at all I mean it is it is the sum of all the integration of all the wave lengths as we as we said as I said earlier for each temperature there is a particular wavelength range within which my body is going to emit it is going to emit this is integrated we will see that how are we going to do that integration before we get to that this is I just want to spend some time because actually there was a paper by I will put this in moodle there was a paper by John in 2007 he has written this paper just to bring what are the contributions of Stefan okay Joseph Stefan his life and legacy in thermal sciences okay I will be putting up this in moodle so it is so beautifully explains although we know Stefan so much we do not know him I mean he is anonymous to us in the sense that we do not really realize his contributions he is Stefan okay so how did he figure out that he has to take t to the power of 4 it so happened that he was the first person to measure the thermal conductivity of gases accurately in all the experimental results which were reported for radiation there were losses but they were not able to quantify the losses because air was sitting inside so they had to minus the conduction they had to deduct the heat loss because of conduction because of the presence of air so for that I need to know the thermal conductivity of air he was the first guy to measure the thermal conductivity of the air properly he gave the number 0.0211 but today's value is 0.02635 that is the value 0.0 he measured 0.0234 okay but today's value is 0.02635 because he could get the thermal conductivity properly he did the curve fitting earlier curve fittings were tried was some constant into some constant to the power of t definitely it did not work okay but he deducted the losses and then he did simple curve fitting on it then he found that it was coming out to be t to the power of 4 how did he check that whether it works well or not somehow he had the data of sun's temperature he could predict the sun's temperature based on this and he found that it is 5800 Kelvin that is how Stefan first gave but then Boltzmann happened to join him as a student for his PhD and I do not know whether as a part of his PhD he did come out this explanation of sigma as 5 point when he did when bold Stefan did the curve fitting it was around 5.15 into 10 to the power of minus 8 Boltzmann derived it and found it as 5.67 into 10 to the power of minus 8 but I do not know really whether Boltzmann did this as a part of PhD work or later work I do not know but Boltzmann continued to stay with Stefan they were both in University of Vienna only and Stefan is known for other problems also actually moving boundary problems for melting someone was asking yesterday melting that is for moving boundary problems also in fact there is a number called Stefan number that is sensible heat upon latent heat. So that ratio is called Stefan number Stefan has not just made contributions in radiation the point I want to strike he has measured he has solved moving boundary problems that is the melting problems he was he was doing solidification of ice how is ice getting solidified in North Pole and South Pole that he had done that is how he landed up with moving boundary problem. So there are plenty of contributions of Stefan and Stefan Stefan was on a personal note Stefan never left his lab he slept in his lab most of the times okay but Boltzmann was quite contrary to that the word used in this paper is peripatetic traveler that is used to travel all around the place but the unfortunate news in all this is that Boltzmann committed suicide committed suicide he did not have a natural death so it is a great loss perhaps he would have contributed much more than what he has contributed if he had not committed suicide but it is believed that he did not get the credit as much as he thought that he should have got at that point of time so that is why he committed suicide that is what people believe but anyway the point is on a philosophical note things are not that easy for even great scientists whom we believe that they are great today okay so that is what I just wanted to I just told all this because we do not this was all done in before 19 19 today's century so we are what we are studying today is done hundreds of years back okay okay so with that I guess we will move on to I will come to this little later how can I imagine again one more controversial and difficult to understand concept let us see whether we can complete this concept how how can I get a perfect black body see I understood that perfect black body is one which emits perfectly and absorbs also perfectly so the visualization is that if I have a cavity if I have a cavity and if I have a small hole in that if I have a small hole in that whatever comes in it undergoes multiple reflections and eventually it has to absorb converse of this also we say it is undergoing multiple reflections while undergoing multiple reflections it has to go out through this hole so that is how we visualize that isothermal cavity with a small opening can be perhaps taken as a perfect black it is true in fact when you calibrate when you calibrate when you when you are calibrating your heat flux sensors for radiation perfect black body is indeed made as a cavity and you put your sensor inside only problem there is in this cavity if I have to make it perfectly radiating what is that I should be doing in that cavity vacuum if there is air and getting perfect vacuum is very difficult you can only get some vacuum you can go to few millibars but after that it becomes whatever vacuum pump you use you are going to end up with small pressure so perfect vacuum getting is difficult so if you cannot make perfect vacuum what is the extraneous thing which gets into natural convection natural convection convection currents are generated then again as I said we cannot deduct we cannot get solely one mode of heat transfer so natural convection is very hard to remove in my cavity if I can generate a cavity in which there is no medium it is a perfect vacuum then I can get the black body is that okay is it convincing so then then rest of the radiation actually we have crossed the rough edges we have crossed the rough route okay so we are now the track is straight forward now we can afford to go fast fine coming back to Planck's distribution this I cannot derive I will have to just state Planck's distribution spectral distribution is intensity now I can understand the intensity lambda, b for a black body is given to be 2 h c0 squared what is c0 velocity of light in vacuum and h is the Planck's constant k is the Boltzmann's constant and t is the absolute temperature one word of caution in radiation as opposed to convection and conduction is temperatures are always supposed to be in Kelvin so I keep telling my students irrespective of which mode you are handling blindly let us use Kelvin because when I am handling two modes then I am in trouble so I have to be very cautious why I have to be cautious let me blindly use Kelvin so on the first day of your class you please tell our unit of temperature is going to be Kelvin no not degree Celsius if we decide on that there are believe me number of errors would be substantially less they will be getting more marks you do not know they have made that mistake you do not know how many marks to deduct now everything else is right so then you are in trouble so that is why let us take that conscious decision okay if I have this Planck's distribution now this is again Planck in fact radiation is full of Nobel prizes okay so there are five Nobel prizes who have which have come from radiation work and everything has occurred independently almost at the same time so Planck was he was Planck and he was awarded Nobel prize for this Planck's distribution okay and I will not spend too much of time and these were the guys who worked with again Helmholtz and Kirchhoff and again Bunsen all of them stalwarts were together only actually okay and it all happened in Europe only you can see one common is generally in fluid mechanics also yesterday we saw German so everything is reasonably coming out from Europe that is why Hitler thought that he is a great guy yeah yeah so so then we have the emissive power if I have to integrate for all the what is that if I have to get the emissive power the Planck's distribution is dependent on only on in this equation it is dependent on what wavelength and temperature if I fix the temperature if I fix the temperature and integrate it for all the wavelengths for all the wavelengths is that right is that right what I mean if I am not I have not integrated all the wavelengths all that I have told I have said this why did I write e lambda lambda comma t equal to pi i lambda comma b lambda comma t what is that I have assumed in the while writing that pi i lambda comma b when can I write that we have derived a little while ago for what it is good that I got confused here and now it has slowed me down pi pi pi where did it come from hi equal to pi I did I how did I come I will have to go to radiation one to show that I thought that is if it is a diffuse emitter then there is no dependency on the direction then I can here it is written a black body is a diffuse emitter that is what we stated in the definition of the black body so I can write emissive power equal to pi i lambda comma b so that is how I end up with e lambda comma b and of course so many constants handling becomes difficult so we have grouped them and made 2 pi h c0 squared at c1 and h c0 by k as c2 so now it looks little hand level hand level or tractable so e lambda comma b lambda comma t equal to c1 this using this what can I do now if I fix my temperature if I fix my temperature I can compute my emissive power over I can I can calculate my emissive power so who decides that for this temperature only this wavelength is coming out who is deciding that by this relation this relation is going to decide whether at a given temperature what is the wavelength range which is going to contribute for my thermal energy radiated or electromagnetic radiation radiated by virtue of temperature when I say by virtue of temperature I mean that in my relation I am fixing my what will be the emissive power at temperature 0 0 Kelvin absolute 0 okay yeah we will go for Chai and come back okay.