Now, this part will be a little bit harder because now we will introduce some of the special relativity. That's one of the masterpiece of Albert Einstein. He was one of the genius. So, the relativity of magnetic and electric fields is a big topic, and when we said that the magnetic force on a charge was proportional to its velocity, you may have wondered, "What velocity? With respect to which reference frame?" Magnetism and electricity are not independent things but should be taken together as one complete electromagnetic field. In other words, these two are just a tool to understand the force interaction between charge. So, one can even appear and disappear depending on which reference frame you are referring to. In nature itself, there's a very intimate relationship between them that arises from the principle of relativity, and let's see what our knowledge of relativity would tell us about magnetic forces if we assume that the relativity principle is applicable to electromagnetism. So, here's a big question mark just next to Melody. So, I hope you will enjoy this journey with us. So let's think about interaction between a current carrying wire, that is this one, and a particle with a charge q, that is that one. Suppose we think about what happens when a negative charge moves with the velocity v nought parallel to a current carrying wire. So, here a negative charge is moving to the right side with a velocity v nought while the wire is fixed. Now, what goes on in two reference frames, S and S prime? Here, S is fixed with respect to the wire, so wire is fixed. The only one that is moving is the negative charge outside a wire, but in fact, there are also carriers that are moving inside the wires because current is flowing. An S prime is fixed with respect to the particle. So this is fixed with respect to the outside external charge that we just mentioned. In that case, the wire itself is moving along with internal charge carriers moving as well. So that's more complicated. Now, in the S frame, there is clearly a magnetic force on the particle. Why? Why do we have magnetic force in S frame? Melody. Because there's velocity times the charge, so we have the force and the velocity of the charge. Exactly. The charge has velocity, and the current carrying wire is creating magnetic field. So qv cross B makes a force, that magnetic force. Now, in the S prime frame, there can be no magnetic force on the particle because its velocity is zero. Even we have magnetic field due to this wire since the particle interest has zero velocity, the qv cross B will be zero. So now you're puzzled. So, depending on whether you are staying with the wire or moving with the particle, you can turn off and on magnetic force. How does that sound? That sounds impossible. That sounds impossible, so let's see. So, as Melody mentioned, intuitively it's hard to accept. It's hard to accept the fact that depending on the observer, whether the observer moves with the charge or not, you can turn on and off the magnetic force. It seems very weird, but this is an extreme case to show you that magnetic field and electric field, they are a whole part of the force and depending on the situation, each one can take different portions. So, let's see. Force acting on the particle in the S frame. Electric currents come from the motion of some of the negative electrons called conduction electrons while the positive nuclear charge and the remainder of electrons stay fixed in the body of the material. We know that. The electron only close to the Fermi level are mobile that are located near the edge of the conduction band for example. Now, we let the density of the conduction electrons be rho and their velocity in S be v. The density of charge at rest in S is rho plus because those positive charge are the nuclear charge that are fixed, which is equal to rho minus in an uncharged wire. In that way, we can keep the electron neutrality in the wire. There is no electric field outside the wire and the force on a moving particle is just the magnetic force because we have the same number of positive charges, the same number of negative charge even though negative charge are moving, it is in steady state, where the charge density doesn't change. So, we know we only have magnetic force which is qv nought cross B, and we know from Ampere's law, the magnetic field from the wire is this one. If we add them, then the force will be this one, one over four pi epsilon nought c square, two times I qv nought over r. So, continuing our discussion on the force acting on the particle in the S frame, as you can see, you have wire with a plus charge that are fixed and with a density of rho plus and velocity will be zero. So v sub plus is zero and electrons are moving to the right side, so the current is flowing to the left side because it's opposite, and it has the same density as the plus rho sub minus and the velocity of v sub minus will be v. We are rewriting the equation that we just derived from Ampere's law and our equation of Lorentz force. If you take a look at I, the current, is nothing more than the rho sub minus v, that is the current density of electrons, times area of this wire, which is A. So rho sub minus A times v will be I. So if I replace that and if we think to make this problem simpler, assuming the velocity of my electrons is the same as the velocity of the particle interests, then we can put v equals v nought which is a special case then v nought becomes v so it becomes v squared. If you take a look at this equation, you may recognize there's some term related to special relativity, v squared over c squared. Indeed you can see the force is q over two pi epsilon nought rho sub minus A over r, v squared over c squared. So, now let's move on to another frame, force acting on the particle in an S prime frame, and this will be a little bit difficult to follow or understand. So, what happens in S prime in which the particle is at rest and the wire is running past with the speed v? The positive charge moving with the wire will make some magnetic field B prime at the particle, but the particle is at rest, so there is no magnetic force on it. So, let me draw this. So, imagine your wire is a bus. It was staying still but now because you are running with the particle or say a person which was outside the bus, if you're running with them, to you, it seems like your bus is moving to the left side and you guys are standing still. When that happens, when all of a sudden in your world, bus starts to move to the left side, then those that were standing still seems to move to left side, and those that were running to the right side with the same velocity, but with opposite direction, seems to be standing still. So, this is the plus charge, so plus charge creates current to make a magnetic field, right? Interestingly, because plus charges is going to the left, the current seems to be going to the left as well. As in the case when you are standing still, when your electrons we're running to the right side because the current is opposite to the flow of your electrons. So, no matter you are standing still or running with your particle of interests, in that wire, you will have current to the left side which will create magnetic field. However, your particle of interests has no velocity, is standing still. So, V cross B will be zero, because V is zero. So, that's why even though you have succeeded in creating a magnetic field, even when running with the particle of interest, we cannot exert a magnetic force on a particle. So, now you may scratch your head, what happens? How can I turn off the magnetic force, and still see the same effect? Still see the same force, right? So, if there's any force on the particle, it must come from electric field. Why? [inaudible] is a multiple choice. From Lawrence force, we hear either electric or magnetic or both. If you lost magnetic, somehow, you have to have electric, right? So, it must come from an electric field, but then you can scratch your head. Why? It is impossible, because you have the same number of charges, plus and minus charge. How can I suddenly create a non-electroneutral case, right? So, it must be that the moving wire has produced an electric field. Somehow, they have disturbed the local electron neutrality. So, it must be that a neutral wire with a current appears to be charged when set in motion. So, let's see if this hypothesis makes sense. So, we will discuss a little bit of a difficult concept like contraction and dilation of space, and also time. So, relativistic contraction of distances, and its impact on charge density. So, charges are always the same moving or not, period. So, the charge Q on a particle is an invariant scalar quantity, independent of the frame of reference, so that's constant. We need only worry about the fact that the volume can change because of the relativistic contraction of distances. So, let me ask Melody. In modern physics, what have you learned about the volume when you have a moving object? Will the moving objects volume change or not? I think it will decrease, right? It will decrease, yes. It will decrease. So, if we take a length L naught of the wire in which there is a charge density rho naught of stationary charged, it will contain the total charge Q is equal to rho naught, L naught, A naught. If the same charges are observed in a different frame to be moving with velocity v, they will all be found in a piece of the material with the shorter lengths. That is to say, the length L is equal to L naught, times square root of one minus V squared, over V squared. Although Vi is very small in comparison to C still, it is subtracting your number that is multiplying L naught. So, you can see lengths is being shorten, but with the same area A naught, since dimension transverse to the motion are unchanged. Good. So, with this knowledge, let's move on to the next slide. So, if we call rho, the density of charge in the frame in which they are moving, the total charge Q will be, rho L times A naught. So, again, if we call rho the density of charge in the frame in which they are moving, the total charge Q will be rho L, A naught. Why is that so important? Because, again when I was moving with the particle of interest, which sign of charge were stationary and which sign of charge are mobile? When you're moving, the plus sides are mobile. Yes. All the electrons are negative. Exactly, so plus charge are mobile, and electrons are stationary, okay? So, that's a very important concept. So, the charge density of moving distribution varies in the same A as the relativistic mass of a particle. So, you can see rho is equal to rho naught, over square root of one minus V squared, over C squared. So, in other words, if the volume decreases, your density increases, okay? So, let's discuss rest density versus mobile density. We now use this general result for the positive charge density, rho sub plus of our wire. These charges are addressed in frame S. In S prime, where the wire moves with the speed v, rho prime plus becomes rho prime sub plus, is equal to rho sub plus over square root, over one minus V squared over C squared. Meaning in S prime, the positive charge density increases, right? The negative charges are at rest in S prime, so that they have their rest density not in this frame, right? In S prime frame. So, rho sub minus will be the mobile density when we're in S prime. So, minus rho will be equal to rho prime sub minus, over square root one minus V squared, over C squared. So, the rho prime sub minus will indeed decrease in S prime, that means plus increases, minus decreases. Then what happens? Will the wire be positively charged or negatively charged? I think it will be positively charged. Positively charged. So, if I have positively charged wire and negative charge outside, what happens? They'll have some attractive force. Attractive force, there you go. Now, let's think about the other case where the negative charge were moving in this direction. So, you have a current to the left, and your wire has electrons moving to the same directions, you have the same current. If you have current in parallel, what type of force do have? Repulsive or attractive? I think attractive. Attractive. Now, you have the same result, but through different tool. For S, you have through magnetic field, for S prime, through electric field. So, now you understand why we have to think both of them. They are only tools to understand the force interaction, okay? So, now we know that we have electric force acting on the particle in S prime frame, and the electric field at a distance r from the axis of a cylinder is E prime, which is equal to rho prime A, over two pi epsilon naught r, which is equal to this equation. Now, the force on the negatively charged particle is toward a wire as we know. The electric force in S prime has the same direction as the magnetic force in S, as Melody just mentioned. The magnitude of the force in S prime is, F prime is equal to, q over two pi epsilon naught, times rho sub plus A over R times V squared over C squared, over square root of one minus V squared over C squared. Now, comparing the results for F prime with our result for F, we see F prime is slightly different, is almost the same if V is close to 0. Then, we can ignore this term, so it's almost F prime is F. But you see still, the finest velocity, F prime is slightly larger than F. So, why is this the case? Right? You may wonder how can force increase as you move around, right? So, we call this relativistic electromagnetism, I hope I convinced you that depending on the frame you're working with, you can either have magnetic field or electric field which will result in the same or similar force, and we see that the magnitude of the force and almost identical from the two points of view, right? For small velocity, we have been considering the force, two forces are equal. Now, we can say that for low velocity at least, we understand that magnetism and electricity are just two ways of looking at the same thing, right? From this equation. But still you may wonder, why are they different? Okay? Why are they different? I think because you have to consider other things like time-. Exactly. So we will just move on to the next conceptual advance where we think about how time changes and how that influenced the force. All right. So we will discuss transverse momentum of the particle and ask you what transverse momentum will the particle have after the force has acted for a little while? We know that the transverse momentum of a particle should be the same in both the S and S prime frames because of the conservation of momentum. Calling the transverse coordinate Y, we want to compare Delta P sub Y and Delta P prime sub Y. Using the relativistically correct equation of motion, F equals dp over dt which means the derivative of momentum as a function of time is force, then we can say Delta P sub Y is F Delta P, Y prime is F prime Delta T prime. So now if you want to equate them, you already now have a hint that F Delta T should be equal to F prime Delta T prime. We have found that we get the same physical result whether we analyze the motion of a particle moving along a wire and a quarter system at rest with respect to wire or a system at rest with respect to the particle. In the first instance, the force was purely magnetic. In the second, it was purely electric although they're still magnetic field because of the current, but it produces no forces under stationary particle. We also know for the moving frame, the time ticks in a different way and would more being more specific, Delta t will be equal to Delta T prime divided by square root of one minus V squared over C squared. So, we will have a larger Delta T when compared with Delta T prime. So that together, because we know the momentum should be equal and if you use that equation, then we now understand why the force was different. The force was different by the exact amount the time became different. So, the electromagnetic interaction of particles is the centerpiece of electrodynamics. If we had chosen still another coordinate system, we would have found a different mixture of E and B fields and electric and magnetic forces are part of one physical phenomenon, the electromagnetic interactions of particles. The separation of this interaction into electric and magnetic parts depends very much on the reference frame chosen for the description and then complete electromagnetic description is invariant. Electricity and magnetism taken together are consistent with Einstein's Relativity. So let me ask my teaching assistant Melody. So in Einstein's special relativity, what was very special about the velocity of light? It is not like the other velocities where it changes depending on your reference. Exactly. Light, velocity is always constant, no matter which reference of frame you're. So here I'm going to defy some myth, so you probably may think if you have a field line around your magnet, let's say this is your magnet and it's creating a field line like this, let me write it in this way. Then if you move your magnet in this direction, you may think this magnetic field line will move together, but this is not the case. So this is the limitation of magnetic field lines. When you move your magnets, you have to draw the field lines all over again. So again, it makes no sense to say when I move a magnet, it takes it's field with it, the lines of B are also move, there's no way to make sense in general out of the idea the speed of a moving field line. The fields are our way of describing what goes on at a point in space. In particular, electric field and B field, tells us about the forces that will act on a moving particle and what is the force on a charge from a moving magnetic field doesn't mean anything precise. So the force given is given by the values of E field and B field at the charge and our mathematical descriptions deals only with the field as a function of x, y, z and t would respect to some inertial frame. So, I want you to keep that in mind. Now, how about electromagnetic wave? We will be speaking of a wave of electric and magnetic fields traveling through space as for instance a light wave, and it is like speaking of a wave traveling on a string. We don't mean that some part of the string is moving in the direction of the wave, we mean that the displacement of the string appears at one place and later another. In electromagnetic wave, the wave travels but the magnitude of the fields change. So in future, when we speaks of a moving field, we should think of it as just as a handy short way of describing a changing field in some circumstances. So here we're going to discuss the transformation of currents and charges using a four vector concept. So it should be noted that charge and current density are the components of a four vector, we have seen that if rho sub zero is the density of charge and the rest frame, then in a frame in which they have the velocity v, the density is rho is equal to rho sub nought over square root of one minus v squared over c squared. In that frame, their current density is j is just rho times v. So it's rho nought v over square root one minus v squared over c squared. Now, we know that the energy u and momentum p of a particle moving with velocity v are given by the equations here below where m nought is at rest mass and we also know that u and p form a relativistic four vector. So, here is a tabular approach of four vector approach. If we wish to transform rho and j to a coordinate system moving with a velocity u in the x direction, we know that they transform just like t and x y z that we learned from special relativity. So you can see on the left column here how x y z and t transforms if you move a frame of reference with velocity v. This is x prime, y prime, z prime, t prime likewise, jx prime, jy prime, jz prime and rho prime will change in a similar fashion. So our special case corresponds to rho equals zero and u equals minus v where j equals rho sub plus v. With this equation, we can relate charge and currents in one frame to those in another. So the result we obtain for the motion of particles, will be the same no matter which frame we choose. Let's discuss about the superposition as well as the right hand rule. So you may remember for magnetostatic especially magnetic field, we use the right hand rule to define the circulation direction of B field around the current carrying wire and we will conclude this chapter by making two further points regarding the subject of magnetostatics. The principle of superposition also applies to magnetic fields because the basic equations for the magnetic field are linear in B field and j. The field produced by two different static currents is the sum of the individual fields from each current acting alone. Physically observable quantities in electromagnetism are not right or left-handed. Electromagnetic interactions are symmetrical under reflection. So, again, we just for convenience use right-hand rule to describe the rotation, sense of rotation for magnetic field but again for a force, when we define v cross b that was again one more time a right-hand rule. If we were to define it by left-hand rule, the result will be the same. So we start with left-hand rule for magnetic field rotation and then with the Lorentz force also left-hand rule, then we will have the same force. It's only a matter of convenience. All right. With that, we'd like to wrap up our lecture here and we'll see you soon. Bye-bye.