Hello, everybody. So, I will start my lectures by summarizing what we did yesterday. So, the first point was that we defined current - and we said current - is a flow of electricity or electric charges. If you like, the ability of a material to conduct electricity depends on the properties of the material so ability to conduct.

In particular, we are interested in what are known as conductors. We observed that conductors have what are known as free electrons. The free electrons belong to the material as a whole and are not bound to a particular atom or atoms. The point about the current electricity is unlike electrostatics, where the field inside a conductor must be equal to zero, the electric field inside a conductor. Having said this, we defined a quantity which is known as charge density. We did point out that electric current itself is not a vector, but we defined current density denoted by vector j, and we said in terms of this. The current is given by j dot ds, where this is defined in such a way that the product, is positive for positive current flow, primarily in conductors the charge carriers, that is, those which are responsible for conduction, are electrons.

However, ah the there are situations where we saw that ions can also conduct, particularly it happens in electrolytes. The point regarding the direction of the current is that, though, electrons are primarily responsible for flow of electricity. The direction of the current is primarily defined as the direction in which positive charges would flow if they were free to flow like the electrons. So, this is direction of the current towards the end of the last lecture we defined what is known as a drift velocity.

We said that drift velocity is the average velocity of the charge flow as a result of the presence of an electric field. So, we found that what happens in the presence of the electric field as these electrons, which are free, they are able to or they will accelerate, and in doing so they will go and collide with the static, ions or atoms and having done that after collision, they would emerge from these ions in arbitrary direction, though, with similar speed with which they had collided, because the collision is nearly elastic.

But, however, since the direction is random, the average drift average velocity of all the electrons taken together will turn out to be zero. But in the presence of an electric field, the there is a general direction in which they move, and that is the average velocity direction we also defined or obtained the relationship between the current density and the drift velocity by this relationship. -e m v d, where this minus sign is because we are talking about the drift velocity of electrons and I have charge of an electron and the number density of the electrons. So, this is the relationship between drift velocity and the current density. Here e is the magnitude of the electronic charge, which is 1.6*10-19 coulomb. Now I will illustrate these with a numerical example. So let me look at a particular problem. Suppose I have a sample of copper having a cross sectional area of 10-7-meter square and suppose this carries a current of 1.5 amperes. We assume that each copper atom provides one electron to the free electron gas and the density of copper. Atoms is rho, so not to confuse with n is 9*103 kilogram per meter. Cube atomic mass of copper is 63.5 units.

So, our problem is to find out what is the drift velocity and the current the current density is fairly straightforward. The current density is simply the current I divided by the area. Current is given to be 1.5 amperes and area is 10-7. So therefore, this is 1.5*107 ampere meter square. I had already obtained this relationship. J=-n e v. So, in order to calculate the drift velocity, I need to obtain the electron density.

The number density of electrons in the sample remember that what is given here is the mass density of the sample, which is 9*103 kg per meter cube now. In doing so, I need to recollect a little bit of chemistry for you, so we know that one mole of copper has a mass of 63.5 grams, which is 63.5*10-3 kg. In addition, I know, and it has number which is known as the Avogadro’s number, namely 6*10-23 number of atoms in one mole with this data, I can immediately find out how many moles are there in one meter cube so number of Moles in 1, meter cube is 9*103 kg/63.5*10-3. Well, I will calculate everything towards the end. So, therefore, the number of atoms there, which will be the same as the number of electrons, is simply this number 9*103/63.5*10-3 multiplied by the Avogadro’s number 6.2*10-10 to the power 23, and this, if you calculate this works out to 8.5*1028 per meter cube and since I have said each atom has one electron contributing to the free electron gas. So therefore, this is also the number of electrons n, so this is also identical to the n that we have talked about. So therefore, my drift velocity magnitude is given by j, divided by n e and j, was found to be equal to 1.5*107, and we have just now found out eight point five into ten to the power twenty eight into electron charge, which Is known to be one point: six introduce power, minus nineteen, all these are in SI units and if you calculate all this, this works out to a rather small number, 1.1*10-3 meter per second, that is equal to 1.1 mm/second, now I would like to compare this number with other types of typical velocities that we have. So, one of the things that you must know is that this is drift speed not to be confused with the speed which the electrons have inside a conductor when it is able to move free. The drift please speed is an average effect of all the electrons taken together. So let us look at what else we can compare this with so average speed of electrons inside the material is fairly large. It’s about 106 meter per second. That is the speed with which the electrons are moving inside the conductor and before they collide, but remember the average of this over because of their directions being random is essentially equal to zero. There is another quantity which I would like to compare this with is the thermal speed of copper atoms. Now this time I am talking about atoms from kinetic theory. You know that the average kinetic energy - let me call it v thermal square - is my equipotential. Partition principle is three by two kt, so thermal velocity of the atoms is of the order of ignoring the factor of 3, and things like that kt over m, where k, occasionally written as kb, is Boltzmann constant, which is 1.38*10-23. In SI unit, which has a little complicated dimension, meter square kg per semen square degree. Now, if you substitute this, what you get is you can see it. This is 1.38 10-23. Let’s take the room temperature to be 300 kelvins, divided by the mass of copper atom, which we just now calculated to be 63.5*10-3, divided by the Avogadro’s number 6.2*1023. And if you calculate all this, it works out to about 2*10-2 meter per second. So, you notice that the drift speed of electrons is even smaller than this thing, and you see what happens if I am talking about electron mass here, because the mass here appears in the denominator this number, that the thermal speed of electron increases substantially and you get This number, which is about 1026 or So, vd, is much less than the thermal speed of both electrons and, of course, even the ions. There is another characteristic speed, and that is the speed with which electric field gets established inside a conductor. Whenever you switch it on now, that is essentially instantaneous, because the electric field speed gets established with a speed of the order of velocity of light. So, therefore drift p speed that we have talked about is an extremely small number. I will come back to the way. The drift speed arises with a little more detail later, but let me now talk about a large class of conductors now. What is found is that a large class of conductors satisfy a rather simple relationship between the drift speed and the current density, and that law, which I will state in a slightly different way, is known as ohm’s law and a very large amount class of conductor satisfies This now, we know that between successive collisions, the electrons are accelerated by an electric field. Now, so, therefore, the drift speed is proportional to the electric field itself and the current density, which is proportional. So let me say that the drift speed is proportional to the electric field, and I know that the current density, by definition or by our derivation is proportional to the drift speed. That tells me that the current density j is proportional to the electric field, and we can write this as j is equal to a constant . This is a vector relationship where the value of sigma is usually large for good conductors and its. This is the property of the material, it’s called conductivity. Now you can see what are the units of conductivity? It is the unit of j/e and j is an ampere per meter square divided by volt per meter. So, it has this unit ampere per volt meter. This quantity is called a semen, so now, normally this relationship is written by writing. The inverse relationship, namely e, is equal to rho j, where clearly, is nothing but one over sigma, and the unit of this is ohm, meter, 1 ohm is same as 1 volt per ampere and is also equal to siemens. Inverse this row like sigma, is independent of electric field, and it depends upon the properties of the material conductors are characterized by high values of sigma or low values of rho. So typical good conductors are, for instance, silver, which has a resistance resistivity of 1.7* 10-8. So, name of row is resistivity, so this is the no meter unit, copper, 1.7*10-8 aluminum, 2.75*10-8, etcetera, etcetera. These are some good conductors. On the other end of the spectrum. There are good insulators. These are conductors. Insulators are those which do not readily conduct electricity. Typically, water is two points. Five into twenty per five. Oh meter, glass can have a value between 10*1014. Between these two classes, there is a class of material which are known as semiconductors, about which you will be learning in detail in one of the reviews of the later lectures now semiconductors are generally insulators at low temperatures and as the temperature rises, their conductivity increases. In addition, the conductivity of semiconductors is substantially affected by impurities that might be present or impurities that might be put into and typical resistivity of certain semiconductors. For example, if you look at resistivity at zero degree of carbon in the form of graphite, it’s of the order of ten to the power minus five ohmmeter germanium is 0.46 ohmmeter silicon 2300 per meter. So, we have talked about conductivity and resistivity, which are properties of the material, but let us now try to talk about a property which depends upon a particular sample. For example, let’s talk about a sample which has a length of l and has an area a cross sectional area A we have seen that .

Now, let’s look at now. I know that if I have a potential difference between the two ends, which is delta v, the electric field is delta phi divided by l, and current density, by definition, is the current that is flowing through it divided by the area a so. Therefore, this quantity is an if you take out the dimensional quantities there, so we will write it as delta v by a delta v by I into area by length. Okay, now we define a quantity called resistance. This is quantity is called the resistance, and this resistance, which is the property of the sample, is given by rho times direct proportionality with l and inverse proportionality with the cross-sectional area.

So, this is characteristic of a sample and, of course, its material. So, you notice that resistance of a sample is directly proportional to its length and is inversely proportional to its cross-sectional area and this r, which we have defined as the applied potential difference divided by the current. If you plot this current as a function of , you find this is essentially a straight line. Now it turns out that a very large class of material follow this simple relationship and in fact, most of the time, unless specifically stated, we assume that the conductors with which you work are ohmic material. So let me sort of ah give you an illustration or an example to calculate ah the resistance of a sample. Let me take a block of copper suppose it has a dimension of one centimeter by one centimeter by 20 centimeter. So let me try to draw it. Not to scale obviously because length should be 20 times as much so. What I do is this now. One of the points that you have to realize is that the response of this to the electric field would depend upon which way you apply the potential difference. So, for example, supposing I decide to apply the potential difference between the long ends? So, this is my l, which is 20 centimeters. Then my resistance between these two ends, which we have said, is rho l divided by the area. Rho is I’ll. Take the data for copper that I had given you earlier 1.3*10-8 and the length is 20*10-2, divided by the area, which is 1 centimeter by 1 centimeter. So, it is 10 to the power minus 4-meter square and if you look at these numbers so here, I have got 2.6*10-5 ohms. Now suppose. Instead, you had applied the potential difference between the rectangular ends. Now your numbers will change now, because what has happened is between rectangular ends, so resistance between rectangular ends. Now I have the same number 1.3*10-8, which is property of a material the length this time is just 1 centimeter. So that’s 10-2 and the area is 20 centimeters by 1 centimeter. So, it is 20*10-4. And if you calculate this, this is 0.65*10-7 ohms. The point to notice that the resist, while you can talk about resistivity of a sample, the resistance, depends upon dimensions and not only that it depends upon. If you want to measure it, then it depends upon where exactly you have applied the potential difference, and so resistance would vary depending upon which pair of points you have applied potential difference. Now, before I close this, let me bring out a similarity between charge, flow and heat flow. Remember that right in the beginning, when I introduced the concept of an electric current, I had brought out a similarity with the flow of water in a tube. Now you will realize that the similarity is lot more striking here and let us again talk about a sample and, let’s suppose that I have a sample of length, delta, x and suppose across this I apply a potential difference of , then, by definition of resistance Of the sample, I know my current, I is , which is times its length , and if you write it remembering that is nothing but . So, I get, so, notice that the current in this situation depends upon the gradient of the potential. So how does potential vary with distance? Now, let us recollect the actually speaking if you wanted to write this as a proper relationship. So, I will write dq/d t, which is my charge flow current, but I will put in a minus sigma, a dv by dx, and that is because the positive charges move in the direction of decreasing potential. So since minus sign, because positive charges move in the direction of decreasing voltage now let us look at what statement I can make about heat flow. If you recall your discussion, when you discussed heat conductivity, you will realize the equation for heat transport was given by dq/dt =-kappa a where this q in this case is actually the amount of heat instead of charge. As we are discussing now, copper is known as thermal conductivity. A is, of course, the cross-sectional area, and this is a temperature gradient and this temperature gradient is required because heat flows from higher temperature to lower temperature. Now we immediately recognize there is a similarity. In fact, the similarity is not just accidental, and there is a reason for this similarity, and that is because transport of heat takes place by transport of electric charge. So typically, a good conductor of electricity is also a good conductor of heat. Let me return back to the discussion of the drift velocity little more in detail. I would be looking at the microscopic aspect of it, but before that, let us remember that we said that vd at most is a few millimeters per second now. This does not imply that, in order to start a current, we will have to wait for long, because it is not as if the electrons are being literally moved from one end of a sample to the other, just like in the case of a water flow. The electrons or the free electrons are already there and the. If you switch on an electric field, the electric field, as we have seen, gets established with the speed of light, which, in this scale, is essentially instantaneous, and it it’s because of this, that you don’t actually have to wait. When you press a switch in your home to light up something because the electrons are already there right up to, let’s suppose you are talking about a bulb, it is all there, and all that you have done is by switching on provide a pushing mechanism, as we Did in case of water, however, there are transients, those that is, the steadiness doesn’t get established immediately, so there is a bit of a time that it takes for the situation to become steady. The second point is that relationship between current density and the drift speed is minus, nav and notice that we have said vd is small. Few millimeters electronic charge is also small. Electronic charge is 10-19. So, the reason why the current density is not that bad is because this is a large number and we had calculated it some time back and we found n is of the order of 1028 per meter cube. So, this number it more than compensate for the product of these two numbers, which is small. So let me now look at. Why is ohm’s law reasonably good, and in order to do that, I will try to give you a microscopic picture of the situation that is happening there. So let me get back to the beginning and we said metals have free electrons and these ones they move. Like a gas inside the material, they do not belong to a particular atom or atoms. What we also said is that these electrons would collide with ions in the material so electrons. I have said already. The typical speed of electrons is of the order of 106 meter per second, so electrons collide with ions and emerge with velocity in random direction emerge from collision now, because the direction in which they emerge from collision is random. If I define an average velocity of electrons inside a material, supposing there is a capital n number of electrons in it and ith electron has a velocity v i so this quantity is 0 on an average. That’s because different electrons are moving in different directions and the randomly, they are moving there. So let us look at what happens if I now put in an electric field in it, so we said in the presence of an electric field, the electrons would get accelerated. So, the way the thing works is this: so, in the presence of electric field so electrons, which I will write for shorthand by e minus get accelerated, but because the sample is full of atoms which are static, they collide after collision. They emerge with velocity direction change once again, they would collide so therefore, this chain acceleration collision acceleration collision. So, this goes on now. What happens is this? So let me try to draw a typical picture. I will show the life of an electron over sometime. Supposing my electron was at this point a I will not show the location of an atom because it will clutter up the figure, but let us suppose I, the electron, went there. So let me also show the direction of the electric field to be given by this. Supposing the electron was directed like that, and it goes there so this is, I will just call it atom number one. It collides there and would emerge from there with a velocity direction. Change, though, there is not much change in its magnitude of the velocity and then, of course, it has a second collision and, let’s suppose it now, this time is directed like this undergoes a third collision. Let’s suppose this time it is, I am trying to draw essentially a random figure, so this is 3, this 4, then it let’s say comes like this. This is 5 and let us say that once again, this is actually an arbitrary direction figure. So do not worry about any pattern in this. This is 6 and then finally, it comes like this, so this is typical. I mean you could draw any way you like in this particular picture. I have shown you. The electron goes through six collisions now. What happens if there is an electric field now, if there is an electric field? Let’s suppose my electron starts from a now remember that the electric field right is in this direction and I have a negatively charged electron. So therefore, the electrons velocity, because it has a velocity in the absence of electric field in this direction, but there is an electric field in this direction, providing it an acceleration that other direction. So therefore, what will happen is that this electron would not quite follow this path, but what will happen is it will take a path which is fairly close to it and maybe go like this now? This path, though, I have shown it as a straight line, is actually slightly curved, though over this length scale, it would appear to be a straight line. The reason is that my direction of the acceleration due to the electric field and the direction of the velocity they are not the same. So, it is very similar to what happens to a projectile when you throw in an arbitrary direction, with the gravity in a particular direction. So, you know that that trajectory is a parabola, but only problem is that in this case my electron velocities are very large, and the electric field that I apply is not that that bad. But what is going to happen as a result is that this path is slightly towards the negative e direction because of the acceleration all right, then it undergoes a collision. There come here second atoms, and then it will go like that, so it will be roughly similar but slightly different notice that instead of arriving at this point. So let me call this original point where it arrived in the presence of electric absence of electric field as b, and this is going to arrive at b Prime. So, there is this slight drift in the direction of minus e okay, and we have seen that the electron velocity is about 106 meter per second and drift. Speed is a few millimeters per second, so that is the electron. Speed is greater than the drift speed by a very large factor. Now let us look at what exactly is the dynamics a little more uh quantitatively, so you notice that in the presence of an electric field, the acceleration of the electron, of course direction we know is opposite - is given by ee/m mass of the electron easy. Now, let us suppose the time between two successive collisions is down. This is also known as relaxation time. That’s where the electron is relaxing after equalism now suppose v i was the velocity of the ith electron immediately after the last time. It collided, then, in time t which is less than tau, because in time tau on an average there would be another collision, but before the next collision occurs the velocity after collision, let’s call it by capital v. So, this is given by the usual formula v. I minus e over m*t- because the I am talking about an electron which must move in a direction opposite to the velocity of the electric field.

So therefore, now remember. I also mentioned that the average value of v i, which is 1 over n times sum over - i v, i = 0. But if you look at this now, then the velocity close to a relaxation time would be VI minus e tau over m, so therefore my average drift speed is average of this, which is of course equal to zero, because this is random but notice that this is not quite random, because it depends upon the direction of the electric field which is given to Be constant so, therefore, this is given by minus e tau.

So, therefore, the magnitude of the drift speed is given by e now this, so this actually connects the drift speed with parameters which depends upon the characteristic things like electric charge: mass. The strength of electric field applied and a parameter which depends upon how frequently the collisions are taking place. So, this is on dynamics of collision, but let us look at the relationship that we have, because we wanted to show why ohm’s law becomes valid. Recalling that my relationship between the current density j and the drift velocity was minus any d that tells me that j is given by n e square tau over m times e, and this has the same structure as equal to sigma times e. Thus, the expression for conductivity is n e square tau over ohm’s law will be valid if sigma remains independent of e, so this means ohm’s laws. Validity is the same as sigma being constant, since in my expression here I have got n e square over m, which depends upon the characteristic there, so this implies. Tau is constant by constant; I mean independent of the electric field. Now this is quite reasonable because we have seen that the electron velocity distribution - okay - this is independent of an electric field and tau, which is the time between two successive collisions, would depend upon the electron, velocity distribution and not on the electric field, and this is the microscopic reason why the ohms law remains reasonably valid. Now I will conclude this with an example. Let me return back to the same example where I worked out. The velocity of the drift velocity was shown to be equal to one point, one into ten powers minus three meter/second and this I we using this. I want to calculate this quantity e m by tau so that my sigma, which is n e square tau over m or inverse relationship, is resistivity. So, resistivity is m/n2 . Remember I have given you the data for , which you have said is one point: 7*10-8. So, therefore, for copper, one point seven into ten to minus eight, which is my resistivity, is m, which we, this mass of the electron so 9*10-31 kg /n, which we calculated in that problem. To be eight points, five into ten to the power: twenty-eight into e square e is one point six, ten to minus nineteen. So, it is two points: five, six into ten to the power minus thirty, eight and this time star. So, take these numbers up and calculate this, and you will find this is of the order of 2.4*10-14 seconds. So, this is a fairly small time during which the electron remains free, and that is the typical relaxation time between two colleges. One occasionally defines a new quantity, known as mean free path mean free path. Is the distance that a typical electron travels before undergoing another column? Now, obviously, since time is , if I multiply this with typical velocity of the electron, so mean free path, which is frequently denoted by lambda or even l, is 2.4*10-14 times. The velocity of the electron, which is 106. Well, let me take 1.66 meter per second typical speed of electrons, and that happens to be about 40 nanometers. So, this is the distance that an electron would move without suffering. Another collision, so, let’s quickly summarize what we did today, the what we did is to look at little more deeply into how drift speeds arise in conductors. The other thing that we talked about is that the drift speed is small, but drift speed being independent.

The drift speed being proportional to the current density and the fact that relaxation time is independent of the electric field is the reason why ohm’s law turns out to be a reasonably good description of the phenomena that happens in case of conductors. We will continue with this in the next lecture and look at certain other parameters which are connected with conduction.

Thank You