 So, we start with the heat flow equation which is q by delta t is equal to conductivity times delta temperature by delta x. We are ignoring the area for a moment because we are going to assume that the area does not vary so that it can be incorporated into the conductivity itself. This is the heat flow equation. We also had another equation earlier that is the specific heat equation. Here we found that the heat input to a substance is directly proportional to its increase in temperature. This proportionality constant depends on the mass of the object and s here is called the specific heat. Now, there are two q's here one and two. There are two delta t's. Here there is a delta x, here there is a delta t. So, let us take a moment to look at these equations and to understand what these different delta t's and delta x's are. First of all this k is called the thermal conductivity. We'll get that out of the way. Now, this q here is the heat flow across an object but this q is not the heat flow across an object. It is the heat flow into an object. They are primarily different because we say in this case that there is heat flowing across an object and we want to look at how that heat moves with temperature change and here there is an object and there is heat flowing into it and this heat causes a temperature increase for the whole object. So, there is heat flow across an object. There is heat flow into an object. An important thing is the difference between the delta t here and the delta t here. Now, this delta t here is the change in temperature or it's not even the change. There is no change here. There is no change. There is a difference instead. There is a difference in temperature between two points on the object, between two points on the object separated by delta x distance. So, there are two points separated by this much distance and the difference in temperature between two points on the same object, on the same object separated by that distance is delta t. But here delta t is the change. There is a change here. Delta t is the change in temperature of the whole object. This is the important difference. Here the whole object is at a single temperature. The different points on the objects are at different temperatures and delta t by delta x is the temperature difference between two points on the object separated by delta x. Here the whole object is at a single temperature and that temperature is changing as heat flows into the object. You can say that this is a change with position, which we call a gradient in physics. And this is a change with time, which we call a rate in physics. So, just like delta t by delta t is the rate, delta t by delta x is the gradient. Before we begin to combine these two equations, we will take a look at one more important difference between these two equations. On the left side, you have q by delta t is equal to k delta t by delta x. And on the right, you have q is equal to ms delta t. What this equation says is that a delta t by delta x, the delta t by delta x, which we call the temperature gradient. Anything, the change in anything divided by position coordinate is called a gradient. Delta t by delta x is the temperature gradient. A temperature gradient causes heat flow. So, it is the right side, this temperature gradient, which causes the left side. But in this equation, heat flow is the agent. Heat flow causes a change in temperature. This equation is in reverse. This one causes this one. A heat flowing into the body causes a change in temperature of the body. There is another very important difference between these two equations. And this leads us to the right method to combine the two equations. You see, this equation says that delta t causes a gradient basically. Delta t by delta x causes q. And this equation says that q causes a delta t. That means when there is a temperature gradient, there will be heat flow from one part to another. There will be heat flow from one part to another. And that heat flow is going to cause a change in temperature of the different parts of the body. If we treat the body as different, made up of different parts, if we treat the body as made up of different parts, then this equation gives us the heat flowing between the parts, flowing between parts. And this equation gives us the increase in temperature as a result of this. So, you can see that there is a sort of loop situation forming here. There is a temperature gradient. There is a temperature gradient which causes a heat flow between parts of the body. That heat flow affects the very temperature gradient which is causing it. The heat flow affects the temperature gradient which is causing it based on the specific heats of the parts of the body. This is our key to combining this equation and generating the heat flow equation. So, before we do that, I will bring these two into the same footing. Here, there is a rate of heat flow across a body which is equal to thermal conductivity times the temperature gradient across the body. We will convert this second equation also into terms of heat flow rate. You know that Q is equal to mass times specific heat times the change in temperatures. If this Q, this heat flow into the body occurs over a time delta T, then the rate of change of temperature, then the rate of change of temperature will be 1 by MS times Q by delta T. So, keeping the agents on the left side and we cause on, sorry, keeping the cause on the left side and the result on the right side, we can write this as, so we keep the result on the left because that is what you calculate and we will keep the cause on the right. So, the cause here is delta T by delta S and the resulting Q by delta T is K times delta T by delta S. Here, we will write that the cause is Q by delta T, a heat flow is the cause and 1 by MS times this cause results in a rate of change of temperature of this. Here, we have that heat flow is again caused itself by a temperature gradient. This is the pair of equations we are going to combine, but we need to do so carefully. First, we will convert it into differential form. So, these deltas, if you take the limit as delta X and delta T tends to 0, we are basically looking at the heat flow in a small amount of time across a small space in the object and we get DQ by DQ because K DQ by DL and we get D temperature by DQ equals 1 by MS DQ by DQ. This is the heat flow equation, pair of heat flow equations through our body. Now, to combine these equations, you have to notice one thing. If we have something like a rod, which is what we are going to consider right now. This equation says that there is a heat flow across the rod. This is 1 and this is 2. Then equation 1 says there is a heat flow across the rod proportional to temperature gradient, which itself implies that the temperature changes with position. T is a function of X, changes with position. And then the second equation says that the temperature of a point depends on the heat inflow to that point. So, that means temperature is changing with time. Temperature is also a function of time because the heats flowing in and flowing out of different parts of the body are not equal. There may be an accumulation of heat at any point of the body and that will raise its temperature according to the second equation. So, first equation says that temperature is a function of X. Second equation says that temperature is a function of time. So, is it a function of X or time? The answer is that it is a function of both. Temperature is a function of position as well as time. The location on the body will tell you what temperature it is and that holds only for one particular time. You need to tell me both which location you are talking about and what time you are talking about for me to tell you the temperature at that point at that time. Now, how would a temperature or any function, how would it depend on two different coordinates? This is something we don't need right now but we are going to use at the end of the lecture that we can say that temperature depends on position and time as a product of two functions. One function is a function of position multiplied by another function of time. So, this is one kind of dependence. Of course, I could have addition here. I could have division here. I could have power or anything. Any affirmative combination of X and T will give me a function that depends on position as well as time. For example, I could say D of X, T is equal to X square plus D square. This is not of this form but this form is a simple kind of form which is something like X square into T square. Here, xi of X is equal to X square and tau of T is equal to T square. When the temperature function depends on position and time as a product of one position function and one time function, then we call it a separable function. Right, so we will come back to this at the end. Here I have just introduced that temperature can depend on position as well as time and if it depends as a separable function our calculations at the end will get much easier. So, we start with these two equations and see how to combine them in a system where there is heat flow across caused by temperature and there is a heat accumulation resulting from this heat flow causing a change in the temperature. Basically, we are going to eliminate Q from this equation and this equation to get how the temperature of the body changes with time. So, the left side equation is about heat flow across a body DQ by DQ by DQ by DQ by DQ. Actually, there should be a minus here because if the temperature is increasing in one direction, heat flow is in the opposite direction. So, now I want you to look at this. This is the heat flow across this part. These are points along the rod. So, this is a point with some particular position X. This is the point with another position X there. So, the difference between them, the distance between them is Dx. Now, there is a DQ by DT here. There is a minus here, DT by Dx here. This is the heat flow in this direction. There is also a minus K DT by Dx here. This is the heat flow in this direction across this line. There is a heat flow in this direction across this line. What is the difference between these heat flows? Let's say it is delta X. What is the difference between these heat flows? What is it? What does it physically mean? So, there is minus K I will take common. It is DT by Dx. How does this DT by Dx change across this gap that will affect the heat flow, right? If I take delta of DT by Dx, the change in temperature gradient. So, I take the change in temperature gradient between these two points separated by delta X and I multiply that with minus K. What I am getting is the heat flow out of the system minus the heat flow into this system. The system being the small region of the rod. So, when I consider this expression, I am subtracting the heat flow out of this region in the rod to the heat flow into this region in the rod. Which means heat flow outflow minus heat inflow which is just the heat accumulation. This heat outflow minus heat inflow which is just accumulation of heat. So, minus K delta of DT by Dx minus K delta of DT by Dx is the accumulation of heat in time DT. In this DT time, in this DT time we consider DQ by DT here. In this DT time, the heat accumulated is such that DQ accumulated by DT is equal to minus K delta DT by Dx. This is part one. Now, what is the effect of heat accumulating on a body? As we just discussed, heat accumulating within a body will cause its temperature to rise according to the equation DQ by DT equal to MS DQ by DQ. Heat accumulating into a body causes its temperature to rise according to this equation. So, then the temperature of this region the temperature of this region is going to rise according to that equation I have just written where the accumulated heat is this. So, we will adopt that equation for this small region DQ accumulated by DT is just minus K delta of DT by Dx. The change in the temperature gradient across the ends of the region when multiplied with the conductivity gives the accumulation of heat. Now, the mass of that small part I can write as lambda times its length where lambda is the linear density. Then I will write the specific heat capacity of the material and I have how the temperature of the body changes with time. Now, I just will take this delta x this side and I get the equation minus K delta of gradient by delta x equal to lambda s into temperature rate. Now, this is very important. This is how the gradient changes with position gradient itself is how the temperature changes with position and now I am talking about the how the gradient itself changes with position basically the gradient of the gradient. If I take limit as delta x tends to zero then this fraction will also become a differential and you will recognize this as the double derivative of the temperature that leaves me with the equation minus K d by dx of DT by dx is equal to lambda s dT by dx and we usually have a location for this. We write it as the double derivative d square T by dx square is equal to lambda s dT by dT. This is the basic heat flow equation. This was studied by great scientist Laplace Boltzmann others. This was studied in 1, 2 and 3 dimensions but in one dimension the equation is very simple, relatively simple where the conductivity times the second derivative of the temperature gradient of the temperature with respect to position is equal to mass density times specific heat times the rate of change of temperature. I want you to remember that this equation is about temperature which changes with position as well as time. There are different temperatures at different positions and all these temperatures are different at different times and we want to now find how is it that this temperature changes with position and time. This is where we use the fact that if the temperature is separable we can solve this equation simply. So I rewrite that equation d square T by dx square is equal to I take A to that sign I will get lambda s by T times this T by dT equation I had. I guess to clear to that side I get lambda s by T. If T is of the form it is a function of position and time but it is a separable function which means I can write it as the product of a function of position with a function of time. If I can write T like this then what will be the second derivative of T with respect to x? Now we are talking only of derivatives with respect to x which means time is not changing then in this case this becomes a constant which means I can take it out of the derivative and I simply have to differentiate this twice and I will get x double dash of this is d square T by dx same thing I will do with time now I need dt by dt and now this becomes a constant because it does not change with time and it is this thing which we have to differentiate but only once in this case when I get x i of x times tau dash of T. What is the use of taking this separable differentiation? The use is that when you substitute back into this equation basically we want to find the temperature function which satisfies this equation what we are going to do is assume that that function which satisfies this equation is separable and then we are going to find these components we are going to find these two components from this equation that will give us the temperature by just by multiplying them we will get back how the temperature depends on position and path. To find these two components of the temperature we will substitute back these equations into this to get now this derivative position was tau of T times by double dash of x is equal to minus lambda x by k and then I have tau of x times tau dash of T. Now we make an important step here you divide by tau times basically take this to this side and take this to this side that will give me tau double dash of x by tau of x is equal to minus lambda s by k times tau dash of T by tau. Now comes the why they are called separable things because if you look at the left side it depends only on x and if you look at the right side it depends only on k so if I change t while keeping x constant the left side is not going to change that means the right side also should not change with time because the left side will not change with time that simply means that this function where tau is differentiated and then divided by itself that function must be a constant otherwise when we change the time the right side is going to change and the left side is not going to change but these two are equal so that means if the right side is not supposed to change with time then tau must be such a function that when we differentiate and divide by itself it gives us a constant that means the differentiation of tau with respect to time divided by tau itself is simply a constant I will call that constant w now this is actually a very easy equation to solve we will get back to the high part in the moment first let's take care of the tau part if I cross multiply that equation I get tau dash of t is equal to w into tau of t and what does this say it simply tells me that tau is equal to that it is a function when you differentiate you get a constant times the function itself which means it must be a into d power omega t that is when when you differentiate this you will get back you can see you can find this by the standard method of solving differential equations you can say 1 by tau times tau dash is equal to omega which means d by dt of log of tau is equal to omega now integrating both sides I get log of tau is equal to omega t which means tau is equal to e power omega t you can solve the equation this way or you can just realize that tau is a function which when you differentiate gives you a constant times the same function back and the only function that satisfies it is the exponential function so now we know the time dependence of temperature if the temperature at any point is t when the temperature will increase or decrease with time as omega times t now we go back to the x part chi of x now we decided that this part is a constant which is equal to omega which means chi double dash of x by chi of x and I will just cross my t line and I get minus lambda s by a times that thing becomes omega and then I had a divided by chi here so it becomes chi this is minus omega lambda s by k times chi of x chi double dash of t you will recognize this equation from simple harmonic function x double dash of t is equal to minus omega square x of t so this means this equation is solved by the simple harmonic solutions which are sine and cosine function which means the chi of x must be equal to some amplitude times cos of the square root of this right cos of square root of omega lambda s by p times x because it is a function of x so now we have solved the two parts of the equation and we assume that temperature is a product of these two parts that means the temperature depends on position and time as we will ignore this constant amplitude the position time dependence is cos of root omega lambda s by 3 times x into e power t this is the temperature time dependence so when you have an equation where temperature depends on position and time and it must depend in such a way that both the heat flow equation and the specific heat equation are satisfied then the temperature must be a function of position and time such that its position dependence is a cosine function and the time dependence is an exponential function graph these functions what we will get is that the rod is like this and the temperature starts off as a cosine function this is the temperature dependence on position and with time what happens is you have an e power omega t term which means this temperature gradient exponentially decreases by time this just becomes something like this which again becomes something like this and it flattens out so the heat flow equation predicts a flattening out of the temperature flattening you can see that the term inside the cosine cos of root omega lambda s by k into x this term increases with omega this means that if omega is higher then this term will be higher which means the cos is going to be more curved if omega is lesser then the cos is going to be less and this has to be multiplied by e power omega t there will be a minus latch this has to be multiplied by this which means more curved temperature more curved temperature false of quicker this is the important prediction of the heat equation that it is not a temperature gradient which matters it is a second gradient of temperature like a curvature of the curve if temperature curves more then it falls off quickly which is very much understandable from a physical standpoint because you want the temperature graph the body wants the temperature graph to flatten out if the temperature has a lot of curvature in it the position difference of temperature has a lot of curvature in it then those parts which are curved will quickly flatten out with time and those parts which are less curved will flatten out more slowly the conclusion of this is that the heat flow equation predicts that body's heat temperature curvature and more curved parts of temperature fall off quickly and less for curved parts of temperature flatten more slowly overall the temperature goes from a curved shape to a flatter shape with time