Good day to everyone. Let's talk about Fourier's Law of Heat Conduction. At a microscopic level of a solid, the atoms look like this. We have the atoms arranged in almost a regular structure called the lattice. This lattice is always in constant vibration as long as the temperature is above zero Kelvin. The more the temperature, the more the vibration. And obviously if the temperature is less, then the vibration is also less. In theory, the atoms stop vibrating only at zero Kelvin. Along with this lattice, we also have free electrons. These electrons always travel in a random direction within the solid. Heat transfer can occur through any of these interactions due to lattice vibration, due to electrons interacting with each other, due to electrons interacting with the lattice. Because the lattice vibration is completely random, heat is able to flow from any direction to any direction. For example, imagine if the lattice vibrates only in one direction, then heat would be able to flow in the direction of vibration and not in any other direction, similarly with electrons. An interesting fact about electronic interactions is if electrons flow in one direction, then electrical current is said to be flowing. Electronic interactions also help in heat transfer. This is the reason why a good electrical conductor is also a good heat conductor. So all in all, conduction happens because of collisions and random motion. Rate of heat conduction depends on several factors. If the area of heat transfer is more, then the rate of heat transfer is also more. This is because more electrons and more atoms can participate in heat transfer. If the length of conductor is more, then heat transfer is less. Because the chance of collision is more, which may send the electrons in random directions, rather than sending it in the desired direction. And finally, heat transfer depends on the temperature difference between objects. If the difference is more, then heat transfer is more vigorous than when the difference is less. To round up, heat transfer is proportional to area of heat transfer and temperature difference, while it is inversely proportional to the length of the conductor. To convert this into an equation, a proportionality constant needs to be put. This is called the coefficient of thermal conductivity. From dimensional analysis, the units of coefficient of thermal conductivity is what per meter Kelvin? If this number is bigger, then you can see from the numerator that heat transfer rate will be more. For better heat conductors, the value is higher. Thermal conductivity of copper is around 380 and aluminum is around 200, whereas for water it is 0.6 and wood is 0.1. There is one last modification to Fourier's law of heat conduction. Heat naturally flows from high temperature to low temperature. It does not flow in the opposite direction, unless before it to happen. It flows until both the bodies reach the same temperature and then stops. Consider the following example. There is a temperature gradient in a body as shown. Let's say the body is made of copper and let's take nominal dimensions. Substituting these values in Fourier's law of heat conduction, we arrive at the answer of course, but more importantly, the value is a negative number. Which means the heat is flowing in the opposite direction. Which means heat is flowing from low temperature to high temperature naturally. This cannot be true. So to work around this problem, we have to place a negative sign in the law. So ultimately, Fourier's law of heat conduction is q is equal to minus k a delta t by delta x. This of course can also be written in differential form q is equal to minus k a doti by dou x. You