 Hi, I'm Zor. Welcome to the New Zor education. Today I would like to talk about a new concept in magnetism. It's called magnetic field flux. Well, let me start from very very simple analogy. We all know that the Sun sends certain amount of energy and if you would like to know how much energy is falling on a specific area, you just have to think about the amount of area in, let's say, square meters. You have to know about intensity of the Sun rays and basically the angle at which it falls on this particular area, right? So that's what basically the magnetic field flux is about, right? So this analogy is a very simple one and I'll probably have some others and in the notes with this lecture, I also have some other analogies. Now this lecture is part of the course called Physics 14's presented on unizor.com. I suggest you to watch this lecture from the website rather than, let's say, from YouTube, where you might have found it because the website besides being free, it also has, for every lecture, it has very detailed notes and also, all lectures are presented as a course, which means there is certain sequence, there is an order, there is an interrelationship between different lectures, so obviously, I'm using certain concepts which I introduced in the previous lectures, etc. So unizor.com is the way to go. You have to choose the course Physics 14 and there is also another course, Mass 14's, which I consider to be a prerequisite for physics because physics uses mass all over the places, like today, for instance, we will use integrals. Okay, so let's go back to our magnetic field flux. So first of all, we have covered what is magnetic field intensity, usually used to letter B, that's the vector. Well, it's like the force, basically, in the field in some way or another. It signifies the strengths of the magnetic field at any point in space where this magnetic field exists. So that's kind of known, it's covered in the previous lectures, so I consider it to be familiar with you. Now, we also know that if you have direct current in some kind of a wire, let's say, then there is a magnetic field around this particular wire. Now, each little piece of this wire creates certain magnetic field around it and next one is also and next one is also. Now, there was a formula for strengths of the magnetic, intensity of the magnetic field if you have an infinitely long wire. There is another formula and we did cover this material before and there is another formula, which was, I think, it was just the previous lecture about current in a loop, what kind of a magnetic field it creates, where I did specify if you have an infinitesimally small piece of a wire, then it creates magnetic field around it, obviously around on the same level here, but also here and here and here and here. So what is the magnetic field intensity created by this particular infinitesimal piece of wire at some point, let's say here? Well, I'll use the differential because it's infinitesimal. So every piece, infinitesimal piece of the wire, it creates the magnetic field of intensity mu zero times i, so mu zero is permeability of the space because it depends on magnetic properties of the space like vacuum is has one permeability and let's say glass has another. i is obviously the current which is running here. Also, obviously the longer this little piece is, the greater magnetic field it creates around it. So we have to multiply it by T s. Now this important distance, obviously the further we are, the smaller the magnetic field intensity should be. So we have to divide it by 4 pi r square. If you remember 4 pi r square, it's the area of a sphere around this particular piece. So the energy which is distributed is distributed onto the whole surface of the sphere, and that's why you have this in the denominator. Now also what's important is this angle. Let's call it alpha, and you have to multiply by sine of alpha. Now why is that? Because if alpha is equal to 90 degree, which means it's it's here. The point is here, relative perpendicular to the wire. It's stronger than on the same distance here because we see wire at the angle. It's actually this part, which is perpendicular, and that actually is smaller. So you have a small triangle here. So instead of hypotenuse, we see the catechus. And that's actually where we're multiplying by sine of A, obvious simple geometry here. So this is a general formula which we have introduced in the previous lecture. Now the next step is the current in a loop. Okay, so let's say we have a loop. Again, every little piece of this wire, every little piece, is the source of magnetic field, all right? So if you will take this piece or another piece, it's the source of magnetic field. So if you have a current here, then each piece creates certain magnetic field around it. Now, in the previous lecture, we have calculated the magnitude in the center of an ideal circle. Now, if it's not ideal circle, well, and if it's not a center even for a circle, it's much more complex. And we didn't really go through calculations. It's just tedious and lengthy technical calculations which don't really bring much knowledge to you. What is important is, however, that at any point, the magnetic field which is created inside this particular loop, inside, I mean the loop considered, the loop is flat. This is XY plane in the system of coordinate. So it's in the plane. This is the plane, this board is the XY plane. Now Z goes perpendicularly. So at any point, since my magnetic field lines are around this wire, so in this plane, all the magnetic field lines will be perpendicular to the plane inside this thing. So within the plane, all magnetic lines are perpendicular. Now, if you go into space, let's say here, then obviously magnetic field lines will be somehow directed at some angle or whatever. But here, magnetic field lines are exactly perpendicular. Why? Well, very simple. Magnetic field lines at any point within this are perpendicular to both the direction of the current, which is tangential to this loop, which means it's lying inside this plane, and the radius vector to this point, which is also lying in the plane because we are only considering the point inside this loop. So the perpendicular to both of them, and both of them belong to the plane XY, obviously it's perpendicular to the entire plane, and the only thing which is perpendicular to XY plane is the direction of the Z axis. So that's why at this particular point the direction will always be perpendicular to this plane. So all the different B's here, here, here, here, here, here, they are all directed into a parallel to the Z plane. Okay, that's good, because if we have, they might not be equal in magnitude at different points, but they have the same direction, that's what's important, which makes our life easier, obviously. Okay, now, in similarity to this concept which I started with, like how much sun energy falls in a specific area, I would like to know how much energy is flowing, magnetic energy, magnetic field energy, if you wish, is flowing through this particular area. Now, it is important for certain practical problems which we will be actually discussing, like self-induction, for instance, etc. So it's a new concept, and it's called magnetic field flux. So I would like to know, I'm using the word energy loosely, it's not really like energy because we're talking about intensity of the magnetic field and area through which all these vectors of magnetic field intensity are going through. Basically, I would like to multiply. I mean, if my field has exactly the same value in magnitude at every point inside this loop, I would just multiply my intensity B by the area of the loop. Now, again, same thing, if sun has all the rays falling on certain flat surface on the earth of the same intensity and actually, for the sun, it's true because it's very, very far. Then we just multiply the intensity of the sunlight measured in some units by the area. The only thing is, if sun is falling at the angle, sun rays, we really should not multiply by this area, we should multiply by this area from here, right? Because only these are really falling here. That's why there is an angle, like in this particular case. So, exactly the same thing we will use here. Now, the only thing is our problem is a little bit easier than in case of sun rays falling at angle because here the angle is 90 degrees always because every intensity vector at any point is perpendicular. So, if I want to calculate the whole thing, I should really calculate it in a very small area. Let's consider this as differential infinitesimally small area around some point P. So, I calculate it in the P. Now, it's perpendicular. So, BP, I multiply by DA by the area around it, differential of the area. So, it's infinitesimally small, like square around it. And that would be my infinitesimal piece of magnetic field flux by definition. So, the definition of the magnetic field flux which is going through the area is for uniform field when all the, at all the points my vector intensity, vectors of intensity are the same. You just multiply the intensity by the area. If it's not uniform like in this particular case, you're talking about differential at very infinitesimally small magnetic field flux which is going through infinitesimally small area DA. Now, why is it infinitesimal? Because inside the infinitesimal area we assume that the field, magnetic field intensity is uniform, obviously. Okay, so that's done and how to calculate intensity of the B. Intensity of the magnetic field at point B. P, sorry, P at point P. Well, that's not easy either because actually every piece of the wire has certain influence on this point. Which means the B, P itself is supposed to be the result of the integration of this along the whole wire. So for every little piece, Ds here, we have to calculate the intensity and then we should integrate it by the whole length of the wire. So that's how in theory it is done. Now, in case my wire is an ideal loop and I'm only calculating in the center, that was easier, that was a previous lecture, if you want to calculate at every point, that's much more difficult and it requires a lot of calculations. But it can be done. But in any case, so our problem right now to calculate the flux is basically two different integrations. Number one, integration along the wire to find at every particular point the intensity of the magnetic field. So we can just have function B of x, y, where x and y are co-ordinate of this point. That's how we get it. And then we have to really integrate it around a circle or whatever the form of the loop is. So that's one thing and that's how we calculate this function. Now B, P means actually B of two co-ordinates where P is the point. And then another integration to get the flux is integration around the whole area. So we have one integral of dB. Another integral would be by dS, B of P times dS. Well, actually it's a double integral because it's error. Now I am kind of loosely using the words integral. I assume that the calculus is familiar to you and concepts of infinitesimal, differential, etc. are familiar. If not, go back to Math 14's course on the same website. I do recommend you to take the calculus chapter at least and know everything whatever is written there because that's what definitely will be used in physics. So these two integration or summarization in a more, I would say, simple language. We have to summarize the combined function, combined intensity if you wish, from each piece of the wire in any particular point and then knowing the function of intensity of the position we have to integrate by area to find the flux. And as a result we get the total flux which is going through this particular area. And again it's kind of equivalent to amount of sun energy falling on an area or for instance you would like to know how much grass is growing in your backyard if you know the rate of growth. So if you know the rate of growth of grass which is kind of equivalent to intensity of the magnetic field then using this area and knowing the whole area of your backyard you can calculate how much grass will go during the season. Now in case of grass we are talking really about uniform rate. Well maybe it's not the case actually but usually you have in the yard you have the same kind of grass and it grows with the same rate but maybe that's not the case so if it's different kind of grass is in different areas of your backyard then you have to calculate separately for each piece and then summarize. That's exactly the same here. You have to calculate separately the intensity produced by each piece of this wire and then summarize because magnetic field as any other field is additive. So if there are like two different sources of the field at any particular point you have to calculate the intensity of one, intensity of another and they can be combined to get the total intensity and we have already covered it in previous lectures. Okay now this is kind of a complex in the general case and it's not even general yet. I will talk about general case. In our practical problems we will usually deal with uniform fields, magnetic fields so the level of intensity is the same at every point and the shape would usually be relatively simple too and it would be probably flat so I mean you can really imagine this wire not to be really like a flat on a XY plane but something like a spiral or something like that and God only knows what kind of magnetic field intensity is produced by this. Now in many practical cases we are talking about loop but many loops done with the same wire in the same place which can be relatively considered to be flat. There is some kind of a volume obviously but you can consider to be flat and that's why you can calculate the intensity of each one, of each loop and then multiply by the number of loops. That's another thing which can be done and we will probably address this as well in some problems. Okay so now I can say that our problems will usually have uniform magnetic fields and probably flat surfaces of the current which is actually creating the magnetic field. Now what else? What's important is the units of measurements. You see the units of measurement comes from here. It's units of magnetic field intensity which is Tesla and units of area which is meter square. So one Tesla falling through one square meter is one Weber. So Weber is a unit. One Weber is equal to one Tesla going through one meter area. By the way from the practical standpoint Weber is a very large unit. Tesla is a very large unit. So usually flux is measured in like mini Teslas or nano Teslas or Weber. So mini Weber or nano Weber whatever. Okay what else? Now let's talk about generality. Again it's just to familiarize yourself with a general concept rather than something which we will be using in real practical problems which I will talk about. So I would like to talk about the flux in much more general case than just this particular loop. Now in the general case is first of all a general field, magnetic field and general surface. Now what is a general magnetic field? Well general magnetic field is a vector function of three coordinates or vector function of a point in a three dimensional space. So if you have a three dimensional space where there is a magnetic field source of which we are not really even discussing whatever the source is. But as a result we have magnetic field. Now how do we characterize magnetic field? Well at any point in the space you have a vector which basically is the intensity of the magnetic field. This is the vector which actually is a source of the force whenever you have another magnet for instance near this. So we know what magnetic field intensity is. General case is that's a vector function of the point in space. That's given. Now general surface. Well general surface in three dimensional space is well general surface and I don't know how to even somehow. This is the general surface. Maybe piece of a sphere, piece of the torus, piece of a surface of a pair, whatever it is, doesn't really matter. So how can we define the magnetic field flux through this surface? Well on the surface, surface is basically part of our space. So for every point on this surface we have a vector function, right? And we would like to draw it in all different directions because I don't know what kind of a source of magnetic field is. But anyway for every point here I have certain value of magnetic field intensity. So what do I do next? Well next I do basically the surface integration that's how it's called. We did not consider it in my calculus course but you probably understand the concept of this. I take infinitesimal piece assuming, well if it's infinitesimal around some point B. I can assume that within this area my magnetic field intensity is uniform. So this particular DA is differential of this area. Now the magnetic field intensity is this. Now in this particular case, in a general case, I don't know if this vector is perpendicular to the surface, right? In my previous example with the loop I knew that my magnetic field intensity vectors are perpendicular to the surface of the loop. And I'm only considering this loop. Now this surface is, God knows what it is, so direction can be any. So I have to multiply by cosine of some angle where angle is basically the angle between the vector of intensity B and normal, perpendicular to this little piece of my surface. So then I have to integrate, it's a double integration for the whole surface, for the whole area A. So I have to summarize. So at every point I have a small differential of the area of the surface around it. And considering my value of the intensity at the point is uniform throughout this infinitesimally small area, I just multiply it by the area and the cosine of the angle between the intensity vector and normal to this particular surface, little piece of surface, perpendicular to this piece of surface. And well, if this angle is zero, which means if my intensity vector B is normal, then cosine of zero will be one. So it's not really participating in this. If my B is tangential to the surface, so the angle between tangential and the normal is 90 degree, I will have zero here. So if my field goes tangential to this little piece of surface, it does not generate any flux. Same thing as with the sun ray. If sun rays go this way and this is the flat area, it does not really warm up this area. It does not really fall on this. It just goes passing it by basically. That's what happens actually around the north pole or south pole. And that's why it's very cold over there because the sun rays are at angle to the surface. And this is a general concept. Again, I just wanted you to have a feel. I'm not really defining anything more precisely than this, but it's a surface integral. So you have to summarize all these little pieces. And there is obviously a certain amount of mathematics which basically tells you how to do it depending obviously on the surface. You can't do it on any surface. You have to do it on some regular surface. For instance, if it's a piece of the sphere and regular magnetic field intensity, which you might actually get. So for any simple, well, mathematically simple cases which can be functionally described, everything is basically calculable. But we are not really getting into this because it's kind of too complex mathematics and not really needed. So we will usually consider uniform intensity fields and usually flat surfaces whenever we will talk about the magnetic flux. But in any case, you have to really know that the flux is a very, very useful thing. We will use it not even in theory, but also some practical usage of the electricity. For instance, transformers. It's all based on the flux. Well, that's it for today. Thank you very much. I do recommend you to read the notes for this lecture on physics for team. You have to go to the course from the unizord.com. You have to choose physics for team, then electromagnetism. And among the electromagnetism, you have properties of magnetic properties of direct current. And one of those is the definition of magnetic flux. So thank you very much and good luck.