Welcome back to Electrodynamics and Its Applications. This will be the last lecture of the entire lecture series. My name is Professor Seungbum Hong and to my right side, I have my teaching assistant, Melodie Glasser. So in this lecture we're going to cover AC circuits. AC stands for alternating current. Let's recap what we learned so far. So we learned the Maxwell equations contain all the static phenomena as well as the phenomena of electromagnetic waves and light, these are dynamic phenomena. They give both phenomena depending on whether r, r is the distance from the source to the point of interest, or point of observation. R is small or large. There is not much interesting to say about the intermediate region, right? And Maxwell equations lead also to solutions which represent confined waves of electric and magnetic fields. In order to lead up to the subject above, we need to cover the properties of electrical circuits at low frequencies. So we're going to cover linear systems with voltages and currents. Melody, when you hear the term linear systems, what comes to your mind? >> So I guess the thing that comes to my mind most is a graph. So if you have your XY coordinate and then you graph your function, so there will be a consistent rate of change throughout the entire system. >> Okay, so let's think this is the IV curve, then linear system really correlates current in the voltage in a linear fashion as Melodie mentioned. Also it can include a super position rule because linear systems are linear in nature. So those will probably come to your mind as well. So here we will deal only with linear systems and with voltage and current which all vary sinusoidally. We can then represent all the voltages and currents by complex numbers, as we learned before, using exponential notation. Therefore, a time-varying voltage, V(t), will be written like V(t) = V caret times exponential to the power of i omega t, where V caret represents a complex number that is independent of time t. It is understood that the actual time-varying voltage V(t) is given by the real part of the complex function on the right hand side of this equation. Similarly, all of other time-varying quantities will be taken to vary sinusoidally at the same frequency omega. So you can see current here is I of t is equal to I caret times e to the i omega t. The epsilon, capital epsilon (t) is emf, electromotive force is equal to epsilon caret times e to the i omega t. Electric field, E(t) = E caret times e to the i omega t. So with that knowledge in mind, we're going to tackle one of the circuit elements that is ideal inductance. An inductance is made by winding many turns of wire in the form of a coil and bringing that two ends out to terminals at some distance from the coil. So you can see the shape of the coil on the right side. So Melodie, when you see this, what comes to your mind? What is the name of this coil? >> So I think this is kind of like a solenoid and it makes that magnetic field kind of like a magnet, as we've discussed in earlier lectures. Exactly, solenoid coil, that's the name of this coil. So we assume that the magnetic field produced by the currents in the coil doesn't spread out strongly all over space and interact with other parts of the circuit. So we assume this magnetic field is confined within this blue box. This is usually arranged by winding the coil in a doughnut-shaped form, or confining the magnetic field by winding the coil on a suitable iron core. Or by placing the coil in some suitable metal box where you have a high-mu metal to shield the magnetic field. Or even better if you have superconducting materials at room temperature, you can probably place superconductors around your coil to shield all the magnetic field. We assume that we can neglect any electrical resistance between a and b. And we assume that we can neglect the amount of electrical charge building up the electric field. So those are the assumptions that you have to think when you see this simplified circuit. Now, let's think about the voltage across an ideal inductance. Suppose we consider the following paths, begin at terminal a here and go along the coil like this, winding coil, staying always inside the wire, to terminal b. And the return from terminal b to a through the air in the space outside the inductance. The line integral of electric field around this closed path can be written as the sum of two parts, as you can see here. This is the line integral of the electric field via coil and outside the coil, all right? From a to b and b to a. Now, the line integral, as you can see here, you have two terms and one of the terms will be zero. Via coil will be zero because conductor has no setup of electric field. We learned that in the very beginning of the lecture, right? So there can be no electric field inside a perfect conductor, the integral from a to b via the coil is 0, and the whole contribution to the line integral of E comes from the path outside the inductance from terminal b to terminal a, okay? Since we assume that there are no B-fields in space outside of the box, this part of the integral is independent of path chosen and we can define the potential of the two terminals. So remember, the current of electric field is a function of the change of magnetic field. But if there's no magnetic field, that means it's zero. Zero, so then it becomes path-independent property. So no matter which way you go, the result will be the same. So the difference of these two potentials is what we call the voltage different, V, as we can see here, and V is equal to minus integral from b to a E dot ds, which is the whole circular integral. And the complete line integral is what we call electromotive force epsilon, and it's equal to the rate of change of the magnetic flux in the coil, because here, you have change in magnetic flux, right? So we have seen that this emf is equal to the negative rate of change of the current, that we have learned. So voltage is equal to minus the electromotive force, which is minus of change of electric current. So that's why minus minus become positive, where L is the inductance of the coil. And since dI/dt is i omega i, for exponential terminology and complex numbers, the voltage is equal to i omega LI. So the way we have described the ideal inductance illustrates the general approach to other ideal circuit elements, usually called lumped elements. And you can see this equation is here. And the properties of the elements are described completely in terms of currents and voltages that appear at the terminals. So you can see, there's a voltage, this is current, and rest of it will be the property of your element. By making suitable approximations, it's possible to ignore the great complexities of the fields. That appear inside the object. And separation is made between what happens inside and what happens outside, as we discussed just before. So for all the circuit elements, we will find a relation like V = i omega L times I, where the voltage is proportional to the current with a proportionality constant. And this complex coefficient of proportionality is called the impedance and is usually written as Z, capital Z. For any lumped element, see V over I = V carat over I carat = Z. And for this particular example, for an inductance, we have Z(inductance) as Z sub L, which is i omega L. And you can see this is an imaginary number, okay? So let's move on to the capacitor. A capacitor consists of a pair of conducting plates from which two wires are brought out to suitable terminals. As you can see here, there are two plates, and you have two terminals here. The plates are often separated by some dielectric material, as we learned in part 2 of our lecture series. And here, we assume the plates and the wires are perfect conductors, their resistance is 0. The insulation between the plates is perfect, so here we have infinite resistance. The conductors are close to each other but far from all others, so all the field lines are confined between the two plates. So all of the fields are confined here, nothing goes out of this plate. This is good for infinite parallel plate. And there are always equal and opposite charges on the two plates, much larger than the charges on the surfaces of the lead-in wires. And there are no B fields close to the capacitor. This is not true, right? We already mentioned that in the past, if we have AC current through the capacitor, it will have change over the electric field, which will cause circulation of B field. But here, we're going to ignore that, and we're going to make assumption that there's no B field close to capacitor. Since there is no magnetic field, the line integral of E around the closed path is zero. The integral can be broken down into three parts, along the wires, between plates, and outside of the box. Here along the wires will be zero. And the integral along the wires is zero because there are no E fields inside perfect conductors, the same reason as we mentioned for inductance. And therefore, the integral will be equal to, the integral outside, here, outside, will be equal to minus of the integral between plates, okay? Since we imagine that the two plates are in some way isolated from the rest of the world, the total charge on the two plates must be zero. So if you see here, the upper plate has +Q, the lower plate has to have -Q. In the very first lecture, we learned about the magnitude of the electric force if the charge balance is not met. So we had an example, two people standing at arm's length, and if there's 1% mismatch of positive, negative charge, what is the force between them, right? So Melody, what was the force? >> So when we calculated that, it actually ended up being kind of the gravitational force of the world, right? >> Yes, the entire Earth. So mass of the Earth times the gravitational constant, that's the magnitude of force that will be exerted between two people standing in a very close distance. So you can now rest assured, if there is a +Q here, you have to have exactly -Q here, so the net charge of the system is zero. The potential difference between the terminals a and b is equal to the potential difference between the plates. So voltage is equal to Q divided by capacitor C. The electric current I entering the capacitor through terminal a and leaving terminal b is equal to the time derivative of charge, dQ over dt, the rate of change of the electric charge on the plates. And if you remember our complex number, and as well as the exponential terminology for sinusoidal wave, then dV over dt is exactly i omega V. And this will be 1 over C, dQ over dt, where dQ over dt is current. Therefore, if we rearrange this, voltage is equal to I over i omega C. So the impedance of a capacitor is Z sub c, which is equal to 1 over i omega C. So Melody, when we learned about inductance, it was an imaginary number. >> Yes. >> We also learned here capacitance is imaginary number. So what does this mean? If you have imaginary number for your impedance, do you think we will have any decay? >> Yes. >> Unfortunately, we will not have any decay. >> Okay. >> It's only when we have real numbers. For pure imaginary numbers, you will never have a decay of current or voltage, so we will learn that. So now we are going to the third element, which is resistor. The third element to consider is a resistor. We will accept as fact that E fields can exist inside real materials, and these E fields give rise to a flow of current electric charge, current. So for rest of the circuit, we assume it's zero resistance. Only here we have non-zero resistance material. And I, the current, is proportional to the integral of the electric field from one end of the conductor to the other. And two wires, perfect conductors, go from the terminals a and b to the two ends of a bar of resistive material. And let's see what happens. The potential difference between the terminals a and b is equal to line integral of external electric field E. And this is equal to the line integral of the electric field through the bar, through the bar, of resistive material. The current I through the resistor is proportional to the terminal voltage V in this type of equation, I is equal to V over R. Do you remember what we call this equation? >> Mm-mm, sorry. >> Okay, probably you want to remember Ohm, which is the unit of resistance, Ohm's law. So Ohm's law states that current and voltage is linearly proportional. And the proportionality constant, depending on how you rearrange this equation, can be 1 over R or R, which is related to the resistance, and R is called the resistance. So we know that the relation between the current and voltage for real conducting materials is only approximately linear. So Melody? >> Yeah? >> Among materials, do you know any materials class that show nonlinear IV curve? >> Yeah, there are a lot of ferroelectric materials, and piezoelectric materials. Almost, actually, every type of material seems like they do. >> Yeah, so ferroelectric materials, piezoelectric materials, or even semiconducting materials. We can have Schottky types of conduction, Poole-Frenkel types of conduction, or even tunneling, right? In that case Most of the IV curve there are non linear. So only a few classes like metals, conductors or even transparent conductors who expect to have these linear relationship. Now, we will see that this approximate proportionality is expected to be independent of the frequency of variation of the current and voltage only if the frequency is not too high. So, before they are being frozen or being relaxed or being, having a resistance, right? So for alternating currents, the voltage across the resistors is in phase with the current, which means the impedance is a real number. So if I have a resistor with this linear relationship, you see it is a real number, and this is the only real number it will have. And because of that, this means, if you flow anything here, you will have a decay of the voltage, okay? Now, so lets learn about the symbols for lumped circuit elements. So this is the schematics of general impedance, where you see impedance is the ratio between voltage and the current. And if it is inductance, this becomes i omega L. If it is capacitance, it's 1 over the i omega C. If it is resistance, it's R. And it can be a combination of those as well. So, results for the three lumped circuit elements, the inductor, the capacitor and the resistor, are summarized in this graph as I just mention. And we have indicated that the voltage by an arrow that is directed from one terminal to another, as you can see from one terminal to another. And if the voltage is positive, if V is larger than zero, that is if the terminal a is at higher potential than terminal b, the arrow indicates the direction of positive voltage drop. So you will have something flowing in this direction, but the voltage will decrease, okay? We can include the special case of circuits with steady currents, by taking the limit as the frequency omega goes to zero. If omega goes to zero, it's almost like DC. So for DC omega is equals to zero. Z sub l is zero, as you can see, omega zero here become makes to zero is short circuit. Z sub c becomes infinity, right? Because omega 0 here, 1 over 0 is infinity, so there is no current if you apply DC here. In fact there's some transient current, but that's just for a moment. And you have Z sub R where R is here only element left when we analyze the circuit for DC. So if you apply DC, this one will only survive, okay? In the circuit elements we have described so far, the current and voltage are proportional to each other, right? If one is zero, so is the either. So an applied voltage is responsible for the current, or a current give rise to a voltage across the terminals. So it's the cause and effect. You have not only correlation between V and I, but you also have causality between V and I, okay? So the elements respond to the applied external conditions, for this reason these elements are called passive elements. So you can now think of, if these are passive elements, then maybe we have active elements, right? And we will learn active elements in the later slides that will be explained to you. So they can be contrasted with the active elements such as the generators, which are the sources of the oscillating currents, or voltages in a circuit. Now, recently, there was a paper in nature that we have a missing circuit element, which is called memristors, so incase recognized, this on is what Melody? >> Resistor >> Yes, and this one? >> That's a capacitor >> And this one? >> Inductor or the solenoid >> Exactly, and we have this funny looking element which is called memristor. So memristor is termed after Leon O Chua, Leon O Chua you can see here. And now we have found some of the resistive change materials behaving like this passive element, to complete all four passive elements in the circuit. Those who are more interested in this fourth element are encouraged to search this paper and read further. Okay, let's talk about an active circuit element, one that is a source of the currents and voltages in a circuit, namely a generator. So, Melody, do you have any experience to play with a generator? >> Yeah, when I was a kid, I had this flashlight with a hand crank. So if you press the hand crank, you can make the light go. >> Exactly, yeah, the similar one was using bicycles, right? To pedal your bicycle or rotate your wheel, and there's a small generator attached to it to light the bulb. But nowadays that's just replaced by LED lamps and battery, because LED is more efficient and the power consumption is much less, right? However, anyhow, supposed that we have a coil like an inductance except that it has very few turns, like here, so that we may neglect the B field of its own current. And let's think this coil sets in a changing B field, as one produced by a rotating magnet, okay? So this is the schematic that we just now described. So, before going further, let's revisit assumptions for generators which we did a few slides before. So, we assume that the magnetic field produced by the currents in the coil doesn't spread out strongly, all over space and interact with other parts of the circuit. We assume that we can neglect any electrical resistance between a and b. We assume that we can neglect the amount of electric charge building up the electric fields. So we're sticking to those assumptions. And we assume that the varying B field is restricted to the definite region in the vicinity of the coil, and doesn't appear outside the generator in the space between the terminals. Now, we consider the line integral of E around the complete loop that starts at terminal a, goes through coil, right, into terminal b, returns to its starting point in space between two terminals, all right? So this is like an airport, you have a tram, right? Connecting two terminals. The potential difference between the terminals is equal to the total line integral of E around the loop. So, voltage is minus the integration of E dot ds. So this line integral is equal to the EMF, the electromotive force in the circuit. So the potential difference, V, across the terminals of the generator is also equal to the rate of change of the magnetic flux linking the coil. Again, we will use the same equation, v = minus epsilon, which is equal to d over dt, d flux over dt, okay? Now for an ideal generator, we assume that the magnetic flux linking the coil is determined by external conditions that is the angular velocity, omega, of a rotating B field. And we assume that magnetic flux is not influenced by the currents through the generator. This is not influencing. So, this generator is not an impedance, so this is not impedance. The potential difference across its terminals is determined by the arbitrarily assigned electromotive force epsilon of t. And you can see the same equation to the right side. So an ideal generator is represented by the symbol shown below. Where you can see here, again, this is the voltage drop. And this is the direction of the electromotive force, which is opposite to the voltage direction. So you can see, the little arrow represents the direction of the emf when it is positive. A positive emf in the generator will produce a voltage V equals epsilon, with the terminal a at a higher potential than terminal b. So this will be higher potential, this will be lower potential. We can also think of an alternative ideal generator. There is another way to make a generator which is quite different on the inside, but which is indistinguishable from the prior one insofar as what happens beyond its terminals. The reason we're thinking of this alternative one is to revisit what we learned in lecture one, where the wire was moving and creating current, or the magnet was moving and creating current. And we were using a different set of equations to explain that. Now suppose a coil of wire is rotated in a fixed magnetic field. A bar magnet indicates the presence of a magnetic field, and connections from the rotating coil are made to the outside world by means of sliding contacts. So these kind of sliding contacts, you can find in many motors, right? Brush, right? And we are interested in the potential difference that appears across the two terminals, a and b. And this is the integral of the E field from terminal to terminal b along a path outside the generator, as usual. And as there are no changing B fields, we may think that there are no E fields anywhere inside the generator. So think about this bar magnet. It's creating a static magnetic field, and nothing should change, right? But if you start to rotate this coil, from the point of view of coil, you have change of flux, right? So that's how this is working. However, we should remember that a total force on any charge inside a perfect conductor must be zero if there are no changing magnetic fields, right? If there are no changing magnetic fields in a perfect conductor, you can not have any force. Because if you have force, it will start to have dynamics that will disrupt the equilibrium, right? So the sum of the electric field E and the cross product of the velocity of the conductor and the magnetic field B, which is the total force on a unit charge, must have the value zero inside the conductor. There's a big if, if there are no changing magnetic fields. So remember, F = E + v x B, and this should be 0 in a perfect conductor, and there is no changing magnetic field. This is called Lorentz equation. And the statement that there's no E field inside a perfect conductor is correct if the velocity of v of the conductor is zero. Otherwise, F on a charge equals zero in a perfect conductor. We see that the line integral of the electric field from terminal a to terminal b through the conducting path of the generator must be equal to the line integral of v x B on the same path. Line integral of v x B along the same path. So therefore, the electric field integration from a to b inside conductor should be minus a to b (v x B) inside conductor. And now you may know where this one is non-zero. So, Melody? >> Yeah. >> Among a lot of parts here, which part do you think will yield non-zero v x B, if you rotate this coil? So it is true that the line integral of E around a complete loop, including the return from b to a outside of the generator must be 0, because there are no changing B fields, right? So the voltage is equal to a to b inside conductor, is equal to minus a to b (v x B) dot ds. Which is equal to minus of electromotive force, which is equal to the d flux over dt. So the line integral of this one is equal to V, the voltage between the two terminals, and minus a to b, (v x B) ds is just the rate of change of the flux linkage through the coil. And therefore, by the flux rule, equal to the emf in the coil. So now you can see the flux here is linked to this rotating part of the coil. And that is the source of the voltage generated inside of this generator. Let's recall. So Faraday's Law to Maxwell's Equation #2, where you learned flux rule using Stoke's theorem, Faraday's law can be written in integral form as follows. We will see the line integral along the closed loop is equal to the minus integral around of round B over round t dot nda, where this one is the flux. So it is the time dependence of the flux that will lead to the appearance of the voltage or electromotive force inside the loop. Also, that's really what we learned through two slides, as you can see. Here this is emf, and this is linked to the time dependence of the flux through the cross-section, okay? All right, also recall in the very first lecture, you had a hanging wire and moving the wire across the standing magnets or moving the magnets across the standing wire. And as I mentioned, for the first case, we use this Lorenz force on the wire. In the second case, we use the flux rule that we just learned. And these will give you the same results, and we can intuitively learn that from the relativity principle. Now, so whether we have a generator where a B field changes near a fixed coil, or one where a coil moves in a fixed B field, the external properties of the generators are the same. So we have gone a long way to reach to this simple conclusion. So let's revisit some of the operation of other generators, chemical cells. So if you hear chemical cells, what types of chemical cells come to your mind, Melody? >> Mostly a battery, but also maybe some sort of biological chemical cell? >> Exactly, biological chemical cells, batteries, or fuel cells, those are all chemical cells. Ordinary chemical cells like battery is a generator, that is to say voltage source, although it will only appear in DC circuits. A chemical cell converts chemical energy into electrical energy. And most batteries are chemical cells. And you may remember from the movie Matrix that Neo was told that we are all but batteries in the matrix system. So we were like biochemical cell in that system. A chemical reaction takes place inside a battery and causes electric current to flow. And the simplest kind of cell to understand is a system composed of two metal plates immersed in a chemical solution. As you can see from the old history, you put two metals inside a vinegar and you create current flow. And in a toy store, you will find you put two metal plates inside apple or orange, and you can create a simple circuit to light up small ball, right? So those kind of things are all chemicals cells. And you see here is the voltaic cell, which is immersed in the electrolyte. And you have zinc and copper, from which one you have reduction, and the other one you have oxidation.
