[MUSIC] Hello, everybody. Welcome back to electrodynamics and its applications. I'm with my teaching assistant, Melodie Glasser, and my name is Professor Seungbum.Hong. So today, we're going to cover the magnetic field, and especially in magnetostatic situation. So the force of an electric charge depends not only on where it is but also on how fast it is moving, right? We learned that from the first lecture when we covered the Lorentz force, which is an experimental equation, found from experiments, and which describes that the force on the charge is depending on electric field and magnetic field. And you can see the electric force is a force component independent of the motion of the charge, so it's only dependent on its position, while the magnetic force depends on the velocity of the charge. So again, the total electromagnetic force, the Lorentz force, on a charge can be written as denoted here, right? So if you look at this picture, see you have a charge here. And if it is moving toward Melodie, to the right direction, and suppose you have magnetic field that is pointing in the right-hand side but a bit up, then the resulting force will be orthogonal to both the velocity and the magnetic field, all right? Okay, so now we have to think about the situation where we have electric currents. So in chapter one we learned about a hanging wire, where we have a wire that carries a current. So I will ask Melodie, when we had that hanging wire and we had a flow of charge through that wire, what happened around that wire? >> So, there's a magnetic field created by the flow of current in the wires, so there's a magnetic field created. And so it will interact with the magnet and the wire will both give force to the magnet and receive force from the magnet. >> Exactly, that's what we learned. So we're going to recap what we learned before step by step and learn some of the terminology that we want to emphasize in this chapter. So let's focus on this question. How can we understand the magnetic forces on wires carrying electric currents? So let's first start with electric current. Electric current is defined by the motion of electrons or other charge with enough drift or flow. And current density is the amount of charge passing per unit area and per unit time through a surface element at right angles to the flow represented by the vector j. So simply put, current density can be thought of as the electric current per unit area, but with a direction, okay? And if we take a small area delta S at a given place in the material, the amount of charge flowing across that area in a unit time is the j, which is the current density, dot n which is the surface normal of the surface of interest, times the area of that surface delta S, where n is the unit vector normal to delta S. So this this one will be the amount of charge. So in other words, this one will be delta q, okay? So in the following slide, we'll discuss how this delta q is related to other parameters we're going to discuss, especially the magnetic field, all right? So in this slide, we will focus on current density j, which is a vector. And the current density is related to the average flow of velocity of the charges. Now, suppose that we have a distribution of charge whose average motion is a drift with a velocity v as depicted on the right side here, so you'd have groups of charge moving with uniform velocity v to the right side to Melodie, right? And when it does happen you can see arbitrary chosen area delta S sweeping a volume. So, here I'm going to ask Melodie when that happens, how much charge will be swept through the delta S inside dead volume. >> Yeah, so if you think about this amorphous shape, it all has the same density. So when we sweep our surface through this volume that is going to be moving at some velocity and some change in time. And so we'll get a volume that's in the shape of a parallelogram and then we can calculate the charges by this equation here. So we know the density is remaining constant and all that's changing is our surface moving at some velocity for some time. And if you multiply these out, you can see that the velocity times the time will give you the distance and then you multiply that by delta S which will give you the volume so density times volume is equal to charge. >> Exactly. So as Melodie just explained to us, the parallelogram has a volume of v delta t, multiplied by delta S. And if you multiply volume by density, you would get the charge. So that's how we get delta q. Now, looking at this equation and comparing with the equation we just discussed one slide before, we now understand the current density j vector is simply rho times v, which is velocity vector. And as you know, rho is N, capital N, times q, so Nqv will be the current density vector, where capital N is the number of charge per unit volume. Okay, so let's move on to current I. We just covered the concept of j, which is a vector. Now current is a scalar. It is a scalar field. And electric current is the total charge passing per unit time through any surface S. As you can see here from the equation, we are doing surface integral of the current density to get our current. So let's take a look at this picture. You have a potato-shaped surface bounded by arbitrary loop and if you think of current density going through this chip then you can evaluate the current through the chip by doing this integration. Now the current I out of a closed surface S represents the rate at which charge leaves the volume V enclosed by S. Because charge can neither be created nor annihilated, which means conservation of charge is kept. Then we can understand as you are picking more charge out of a out of a jar, then you will have less and less charge left inside. And the rate at which you are pulling out the charge will be the curve. >> So one of the basic laws of physics is electric charge is indestructible and is conserved. In another words, in any closed surface, the outward current which is the flux of the current is equal to the rate of the change. The rate of decrease of your charge inside that volume. And as you know, this can be transformed into diversions of the current density. This is the Gauss's Theorem. So if we revisit our Gauss's Theorem or diversion's law, j vector dot n dS will be equal to, this one will be equal to divergence of j. Volume integral and this is equal to, on the right side, we have -dQ over dt which can be rewritten as minus t over dt integral of rule dV. Because this is charge density, this is volume integral. So it will give you the total charge. Now if you switch the integration with the differentiation, then it becomes integral round rural over round t dV. So now what we understand from these equations is this guy and including this one will be the same. So that's what's written on the right side here. So this will be the differential formula to write the conservation of charge. So we are going to discuss the magnetic force on a current. Melody, explained to us that if we have current-carrying wire, it creates a magnetic field and will interact with the magnets underneath that wire and it will give force and take force. It will act and be reactant. So we will understand that from some different point of view. So let's think about the magnetic force and a current. So the total magnetic force delta F under volume delta V is the sum of the forces on the individual charges. That is to say, delta F which is force is capital N delta V times q times v, this is velocity, cross product and B and we know that this will be the number of charge. Number of charge carriers and q will be the charge quantity each carrier carries. So this will be the total charge and this will be the velocity, and this will be the magnetic field. And we know this, the N qv is j. We just derived that. So if we rewrite that, it's j x B times delta V. And if we decompose delta V which is the volume of this wire into the cross section delta A, the area and the length of the wire delta L, then it is A delta L and j x A is current. So I x B times delta L. So this equation gives the important result that the magnetic force on a wire due to the movement of the charges in it depends only on the total current and not on the amount of charge carried by each particle or even inside. So let me discuss briefly with Melody in which case we can have the same current, but with different amount of charge per each particle. So Melody, can you think of any examples where say, ions or electrons carry different amount of number of charges? >> Is there maybe doping? >> Exactly. For example, if we have aluminum doping or aluminum ions, aluminum has three plus. So when aluminum is moving, it is carrying three charges at the same time. However, no matter which carriers you have, what really matters flows down to total current. So that's the most important part here. So it doesn't matter whether electrons are flowing or ions are flowing, or ions with different balance are flowing. Good. Now, let's think about the magnetic field of steady currents and that will lead us to Ampere's law. We were discussing elatostatics where all the charge are fixed in position and time. Now we move that into a bit complex word where the charge start to move, but in a fashion that they are coordinated. So at any time, the charge density is constant. One example you can think of is when all the charges are hand in hand and they are circulating in a circle. So that will be like a steady state where you have a movement, but the net amount of charge at any given time at any position will not change. So we have seen that there's a force on a wire in the presence of a magnetic field produced say by a magnet and we might expect that there should be a force on the source of the magnetic field, that is to say on the magnet when there is a current in a wire. So the wire itself generates a magnetic field that we know. So moving charge produce a magnetic field. So up to here and now, we can ask, given a current, what magnetic field does it make? So when we have a current carrying wire, we know it makes a magnetic field to interact with the magnet. But now, we want to know quantitatively by numbers. So we are now going to use Maxwell's third and fourth equations to solve that problem. So here, we're going to think about magnetostatics where the world is different from electrostatics. So Melody, how does a world of electrostatics look like? >> In electrostatics, there are no charges moving. >> There are no charges moving. Meaning, every charge has its own position and they stay still. We are going to upgrade that world to magnetostatics where everybody will move, but in a coordinate fashion, so that the average density charge will not change. So we call that steady state. Now if we drop the terms involving time derivatives in the Maxwell's equations, we get the equations of magnetostatics, particularly for the third and the fourth equations. And you can see, we drop out the time derivative of electric field. Because electric field will not change if there is no change in charge density and we already know this is true not only for statics, but also for dynamics. So we will have these two equations to start with. Now as mentioned, magnestostatics is a special kind of dynamic situation with large number of changes in motion which we can approximate by a steady flow of charge and this is the study of steady currents. And since the divergence of the curl of any vector is zero. So you can see the j is a curl of magnetic field. And divergence of any curl of a vector is zero, we can already know the divergence of the contents that should be zero, all right? And we know from the conservation of charge, the divergence of j should be equal to minus delta rho over delta t. Which means if this is zero, the charge density will not change as a function of time. Which means charge density is constant and therefore, this is consistent with constant electric field, okay? So let's study Ampere's law. Ampere is a scientist whose name appears in the unit of current, right? So the requirement that the diversion of current density is 0 means that we may only have charge with flow in path that close back on themselves like circuits. Including capacitors that are charging or discharging. So we have to have charge hand in hand in a circle. And that's the essence of circuits, right? And let's take a look at this circulation of the field. And if we take an integration this is line integral of B field along the loop comma. Then Stokes theorem, we can change it to the curl of b surface integral, right? And we know this term now is related to the current density in magnetostatics. So now we can replace this with a simpler form. And if you take a look at this this is nothing but a current. So we can replace this by current through the surface S here, okay? So the circulation of the magnetic field around a current-carrying wire can be calculated using this Ampere's law. This law called Ampere's law plays the same role In magnetostatics that Gauss' law playing electrostatics, meaning it's very important. Very, very important. So in this slide, we're going to think about the magnetic field of a straight wire in a solenoid and then we will also discuss about atomic currents. So let's take a look at this wire, where you have a cylindrical cross section. And now I'm going to ask you, what is the field outside a long straight wire with a cylindrical cross section? Let me first ask Melody about her thought. >> So we saw something similar with the electric field. >> Mm-hm. >> So I'm going to say that with increasing radius, the magnitude of the electric field will decrease. >> That's an excellent guess. So let's see if her guess is right. So first let's do make some assumptions. So let's assume the field lines of B go around the wire in closed circles, which is intuitively right, if we think about the fourth law, where you have a curl around a wire, Ampere's Law. And from the symmetry of the problem, we can think B has the same magnitude at all points on a circle concentric with the wire, because we have cylindrical symmetry. So you can think of the circulation of a B around a circle to be equal to B times 2 pi r, which is the peripheral of that circle. Now, this B times 2 pi r, according to Ampere's Law, is equal to the current through that surface enclosed by the circle that we are thinking about, which is just a current through the wire. So we just put the i there, and put the constance underneath. Then we come with an equation that says the magnetic field is nothing but 1 over 4 pi c squared times 2 times i over r. So as Melody mentioned, the magnitude of the magnetic field will decrease as you go further and further away from the wire and specifically, it's 1 over r law. And in vector form, you can see, it's 1 over 4 pi epsilon naught c squared, times 2 i cross product, the unit vector of the position vector over r. So that will be the exact solution for the magnetic field around a current carrying wire. Now, we are going to think about a little bit more complex structure, which is called solenoid. It is similar to a spring, but you can see a lot of electromagnets using this concept. So, since a current produces a magnetic field, it will exert force on a nearby wire which is also carrying a current. And if the wires are parallel as in the case in this cross section of your solenoid, then each is at right angles to B field of the other. And when currents are in the same direction the wires attract when the currents are moving in opposite directions, the wires repel. Suppose we have a long coil of wire wound in a tight spiral, which is called a solenoid. And according to experimental observation, the field outside is very small compared with the field inside when a solenoid is very long compared with its diameter. So we can assume to its extremity if it is an infinitely long solenoid. We don't have any field outside. All the fields are confined inside. And this is cross-section cut through your eggshell direction. And this is in the flow interaction, okay? Now, so let's calculate the magnetic field inside a solenoid. Since the field stays inside and has zero divergence, its lines must go along parallel to the axis. And that being the case, we can use the Ampere's law with the rectangular curve gamma. So here you have a rectangular loop here, right? And here, we've defined some terminologies. Small n is capital N/L where capital N is the total number of turns. Capital L is the total length of your solenoids. So if you divide N by L it will be turns per length, so it will be density of your turn, right? And if you think about this circulation of B field, as you have no B field outside, the only one that counts will be the one inside here. So B naught times L will be the only one contributing to the circulation of this integral. So B naught L is equal to the current through this surface bound by gamma, right? And you can see a lot of wires here, so that wire's number will be nothing but N times N, right? Capital N, for the length of L. So N times I over epsilon naught c squared. And if I divide n by L, this will be the turn density, right? So B naught is nothing but n I over epsilon naught c squared. So Melody, if I want to make a strong electromagnet, what should I do? Which parameter can I use here? >> I think you can increase the number of turns. >> Mm-hm. >> Or you could decrease the permativity somehow if that were possible. >> Yeah. But as you can see, this is the permativity vacuum. >> Yeah. >> So probably is a hard choice. Maybe it's better to increase the n or increase the current i. >> Right >> But if I increase current i what happens? What happens if I increase a lot of current. What happens to your heater if you- >> Maybe it'll break. >> Yeah, it will burn, right? It will burn. It's a fire hazard, right? So that's why a lot of transformer, when they are overheated they just burn, right? It's very dangerous. So we just learned the magnetic field of a solenoid, and now we're going to tell you that the magnetic field out of the solenoid is equivalent to a bar of magnet, okay? So what happens to lines of B when they get to the end of a solenoid? And the answer is in this picture, as you know, they are escaping out. They're confined inside, and then they're escaping out, and making a field line that is analogous to that of a bar of magnet. Or analogous to an electric dipole. And presumably, they spread out in some way and return to enter the solenoid at the outer end like a magnet. And our theory says that the magnetic effects of iron come from some internal currents. In solenoids give us some hint about the feature of the structure, all right? Here the key feature is rotating current, right? So somehow, all magnetism is produced from currents, in a permanent magnet there are permanent internal currents. In a bar magnet, large numbers of electrons all spin in the same direction and their total effect is equivalent to a current circulating on a surface of the bar. And this is analogous to a uniformly polarized dielectric equivalent to a distribution of charge on a surface. And then in later chapters, we will show you, and prove it mathematically, okay?
