 Hi, I'm Zor. Welcome to a new Zor education. So today I would like to talk about equations which are basically the fundamental laws of electromagnetic field. Well, first of all I would like to actually give some kind of a credit to a very important man in this particular part of the physics, James Maxwell. And there are Maxwell equations. Basically these are four fundamental equations which are describing the electromagnetic field. Well, in as much as three laws of Newton are describing mechanics, kinematics, dynamics, etc. So it's very, very important. And these equations are very important not only for electromagnetic field description, but also they gave the push to the whole contemporary physics, basically, which includes relativity, even quantum mechanics. So I'm not going to present these four fundamental Maxwell equations in their total complexity with proofs, etc., etc. However, I will try to give you, first of all, the feeling of all these equations are all about and, well, derive certain equations in certain simple cases. So that just gives you a flavor of whatever we are talking about. So today I will talk about one particular equation, the first equation, which actually kind of, well, it goes from the Coulomb law, which we all know about electrostatics. So we're talking about only electricity, no magnetism, just electricity, and we're talking about electric charges and what kind of electric field they create. Okay, so first of all, let's talk about Coulomb's law. That law states the following. If you have two charges, q1 and q2, and they are at certain distance r, then the force would be proportional to product of these charges. Now the force can be either attracting if these charges are of opposite sign or repealing if they are of the same, positive and positive or negative and negative. So these are basically charges themselves. Now we are talking about the electric field. So this is something like a force on a distance, because there is nothing in between these two charges, which actually like a spring pulls them together or repels them. So this is something related to a concept of field, which we were talking about before. It's basically part of the space where certain forces are acting, and the sources of these forces are not immediately touching the objects of these forces. So right now we're talking about a particular force which is, well, created, if you wish, by any charge. So if there is some kind of a charge, a single charge, it creates the force which acts on everything around it, everything electrically charged around it. Now we have certain characteristics of the field which basically describes what kind of a force this is. It's called electric field intensity. So electric field intensity would be proportional, obviously, to this particular q. And it's also this same constant, k, and it will be inversely proportional to distance. And that's how we describe something which is called intensity of the electric field. So if you want to know the force which acts on a particular charge, you do this. Now these are two different, this is a source charge, and this is some kind of an object charge. All right? Well, I think I prefer zero to have here, and here without any signs at all. So this is the source of the field. R is the distance from the source of the field to a particular point charge. We're talking about point charges right now. And then the force can be described as this one. So you have intensity of the field times charge. So intensity is basically the force which acts on a unit charge. That's why when you multiply it by q you will get the force. Now what's important is these are vectors. Now direction of these vectors in case of point charges is always radial from a point to a point. Okay, great. So we have defined the intensity of the field which is basically the most important characteristic of any field. In this case we are talking about electric field. So intensity is basically the force which can act on a unit charge. Now what if there is no visible source of electric field? Is it possible that electric field is created but we don't really know about what exactly is the source of this? For instance, the source can be two different charges and we just don't see them. We only feel the source somewhere out there which is on the distance from these two charges. What's important is again to characterize the field by intensity vector. I don't even know about this but if I can measure the intensity at each particular point of the space. So now this is the space and at each point there is something, some vector which is actually a force on a unit charge. So if I take a unit charge and put it here it will be either attracted or repelled to a certain direction which means I can measure this particular force at any particular point. So that's very important. So now we forget about the source of the field, this initial. And we are talking about only about the field itself, piece of space, whatever part of the space around us. And at each point there is some kind of vector which is directed into some direction and it has certain magnitude. So if I have electric charge and I know this vector I basically know the force. The magnitude will be multiplied by the charge and the direction will be whatever the direction of intensity is. Okay, now this is in many cases represented as, so this constant 1 over 4 pi epsilon is k where epsilon is a characteristic of basically a space where exactly all these electric fields are, where they exist. It can be vacuum, it can be water, it can be glass, it can be anything. So the most important part of this case is that I will probably use this particular usage. So instead of k I will use this. Now epsilon has, it's basically some meaning, it's called permittivity. It's how permissive the space is for electric field. It doesn't really matter, this is a constant, this is a constant. This is a constant for any kind of a substance. There is a k and correspondingly epsilon for a vacuum, for air, for anything. So just a constant, but this is more convenient constant as you will see in a couple of minutes. Now we are talking about a field produced by a point charge. Now this field would be obviously radial either repelling or attracting depending on signs of electricity in the initial charge and in the charge which is our probe charge if you wish. So I would like to introduce a new concept. I don't really remember if I introduced this concept before when we were talking about electromagnetism which was the previous part of the course before the waves, I was talking about electromagnetism and its properties. There is a concept which is called flux. Actually it's for any field, not only the central field but for any field. So again we are talking about space and at every point we have some kind of a vector which is a direction of the force and the magnitude of this vector it defines basically the intensity of the field. So the concept of flux, in this case electric field flux, here what it is. If you have certain very small piece of space, a piece of surface, let's say you have a sphere for instance you can have any piece of that sphere that we are talking about or you can have a cube and let's talk about piece of that cube on the side or even in the angle. It doesn't really matter what kind of a surface this is as long as it's compact basically. Now there are certain vectors of intensity which are defined on that particular surface, right? So if this is a surface somewhere in space, since we are talking about space which is basically a field which means at every point in space there is a vector of intensity. There is a vector of intensity at every point here directed somehow, not necessarily radially or whatever. So any surface and any vectors which are basically emitting from it, well it's not really emitting it's probably since it's a field they are going through this particular surface and that's why it's called the flux. Now what's important is that in a very small piece of that surface, obviously infinitesimal is small if we are talking about the real scientific kind of approach, we can consider the vectors to be constant within this infinitesimal piece of the surface and what we can do is we can multiply the vector which is like one and the same vector everywhere times the area of this particular small piece. And that's how we will have an infinitesimal flux going through this particular piece. Then we can summarize the flux through all these pieces and as these pieces, sizes of these pieces are going down infinitesimally to zero, our flux becomes more and more precise. This is basically the same thing as integration but we used to have integration of the function. We divide it into small pieces and then multiply the widths of the piece by the area of the function and then as the piece is going down the whole thing becomes integral. Now this is called a surface integral basically but we don't really care about how it's called scientifically and we will not be talking about very complicated surfaces. The idea is to break the surfaces infinitesimal pieces, multiply, consider that the vector of intensity is exactly the same within each particular piece, multiply the area of the piece by vector and that's how we get this particular intensity. Now I did not specify one little thing. The vector can go from this surface either perpendicularly to the surface or at the angle. Now if it goes at the angle it's not exactly the same thing. What's important is to have a normal which is perpendicular to the surface and project the vector of intensity onto this normal. So if you have a vector of intensity, I think I have exhausted this, if you have a vector of intensity, this is a very small infinitesimal piece of surface. This is a normal perpendicular to the surface and if this is the intensity vector I have to project it and then multiply the magnitude of this projection by this because as we know let's say some which goes perpendicular to the surface hits it much better than if it's under angle, right? So my definition of the flux is the same as I just mentioned before. You multiply the area but not exactly by the magnitude of the intensity vector of this particular in the immediate neighborhood of this point but to a projection perpendicular to the surface. Now what is perpendicular to the surface? Well if it's some kind of a tricky surface perpendicular to any point is perpendicular to a tangential plane, right? So if you have let's say a sphere, what is perpendicular to a sphere? Well you have a tangential plane to a sphere at point, whatever the point is and perpendicular to a tangential plane would be perpendicular to a sphere which happened to be a radial line because every radius is really perpendicular to the whole surface in the point where it points and that's mean it's all perpendicular and that's exactly what I would like to use right now for a simple case but I would like to do, I would like to calculate the flux which is produced by a point charge through the sphere and the point charge obviously is in the very middle of that sphere in the center. Now in the beginning of this lecture I was talking about basically talking about Maxwell equations in some simple cases so this is a simple case and I would like to demonstrate what exactly the first Maxwell equation means for that particular case. Okay, so let's just do it. Let's say we have a sphere in the center we have a charge q0 the radius of the sphere is let's say r now since this is a centrally symmetrical figure obviously all the forces are going radial okay, so this is the force, this is the force etc. they're all radial now I'm talking about the surface of the sphere which means all the points are at the same distance so r is exactly the same everywhere on the sphere and I would like to know what is my flux produced by the field with the q0 as a source through this sphere of the radius r. Well, okay, let's go back and remember the intensity is equal okay, this is my intensity on the distance r from my charge where epsilon is permittivity of whatever the environment we are in maybe it's a vacuum, maybe it's air, it doesn't really matter whatever epsilon is. Okay, now what's important here is that at any point on the sphere my intensity vector has exactly the same magnitude because r is the same and q0 is obviously the same also and every vector is perpendicular to the surface of the sphere this is exactly the simple case because if I will take for instance a cube have a source of electricity here then one vector to this direction will cross the cube and it will be perpendicular but this vector will not be perpendicular and since I'm talking about the flux and the flux basically is the flux on a very small piece here would be an area of this time's projection of the vector intensity onto the perpendicular to a surface, normal to a surface then that's a difficult kind of point which we can do as exercise but integration would be really difficult here integration is trivial, why? because any little piece if I will take here will have already perpendicular force I don't have to project it to anything because the force exactly is along the perpendicular to a surface force is always normal to a surface as we are saying and all the forces on this sphere are exactly the same in magnitude so they are all perpendicular to a surface and they are all the same in magnitude so there is no sense for me to break this surface into infinitesimal pieces and calculate for every piece and sum them up together why? because if I will sum them up together the magnitude would be actually it would be like E1 times S1 plus E2 times S2 but all E's are the same and they are all equal to the E so we take E outside of the brackets of the parenthesis and S1 plus S2 plus etc. will give me the total area of the sphere right? so that's why this is a simple case and that's why I'm bringing it today so to calculate the flux for the sphere in this case it's very simple I take the magnitude of the intensity vector and I know it's perpendicular to every piece of surface and multiply it by area of the sphere which is 4 pi R2 and this is my flux it's a Greek letter phi capital by the way so this is electric field flux so I will put F phi E and well let's see what it is E is as I said so what happens? my 4 pi R2 would cancel out and I will have Q0 divided by epsilon now why is this remarkable? it's remarkable because it does not depend on the radius so the flux through this surface or through this surface or through any surface of the sphere no matter how big the sphere is is exactly the same and this is quite remarkable now let me give you another more remarkable case what if Q0 is not in the center of the sphere? well what we can do we can always I have not done it in Reichenkin in the notes for this lecture it might be as an exercise but in theory if it's not a center I can always put some kind of a diameter through a center of the sphere and through a point where Q0 is located and sum up these two things small ones but this is a smaller this is bigger which means this particular radius well distance to a sphere is less than this one so I will have R1 in one case and R2 in another case however what's interesting is that if I will start summarizing that thing in the way how it's supposed to be with kind of surface integration so to speak I will have the same result what's another important thing is that if I instead of a sphere will take a cube which I was just drawing before and I will just try to calculate the flux through this particular cube surface through the cube I will have the same result so what's interesting is that for any closed surface whether it's a sphere or ellipsoid or cube or parallelogram I mean parallelepiped or whatever else you can come up with even absolutely not geometrically known kind of form as long as it's closed completely from all the sides you will have exactly the same result that the total flux would be equal to this particular thing the amount of electricity inside of that particular point charge divided by epsilon which is a characteristic of the space more than that if instead of one point charge you will have 25 you will have exactly the same result this will be just a sum of all these charges so no matter how charges are distributed inside that surface which is completely enclosing the piece of space you will have exactly the same result the flux from all these charges through this surface would be this regardless of the shape of the surface regardless of how big it is how the charges are distributed inside etc and that's a very important characteristic of electric field and this is something which is essentially the first Maxwell's equation now what I will do right now I will try to put that Maxwell's Maxwell's equation instead of this form which is kind of integrated form into differentiating form and that would probably seem to be maybe a little bit more involved mathematically but since it will be differential which means it will be very small well infinitesimally small pieces when we are talking about differential you will have probably better understanding why we have something like one and the same kind of a law for any kind of shape etc because whenever we are talking about small pieces it's not really a shape which is important and in a differential level the equation would actually mean whatever I am going to derive right now and then we will see how it looks on a differential level ok so for this we will do the following we will consider a very small piece of space and I will use the cubicle piece of space these are coordinates x, y, z and the cube is somewhere in space and let's consider that this cube is aligned with the coordinate plates so this point A will have x, y and z coordinates and the dimensions of the cube would be delta x delta y and delta z well delta are small well again obviously you understand that it will be eventually infinitesimally small but because they are small I can actually consider the field to be more well uniform so to speak because it's not really changing much ok fine so how can I calculate the flux from this particular field well let's consider that I have certain distribution of electric charge inside that little cube there is something which is called density of electric charge and I call it now I assume that this density of electric charge is such that the total electric charge in this cube I will put delta x, delta y and delta z is equal to times the volume so the volume of this cube is delta x times delta y times delta z and if my density of electric charge is this then I assume that inside of that cube I have so much electricity ok so it depends on obviously x, y and z and dimensions of this cube well I shouldn't really say it's cube because I did not assume that delta x is equal to delta y is parallel to the pipette fine it's just too long a word so if I will call it cube sometimes and you understand that this is parallel to the pipette alright now so I know that the charges are inside this is the total number of charges inside ok now I do know basically that as I was saying before the charges are distributed the total flux should be the same right but anyway I'm not right now talking about this now we have to talk about intensity vector produced by these charges alright intensity vector is E of x, y and z and again this is a vector so at any point inside of that cube parallel to the pipette there is a vector ok great how can I find out the flux which is basically I have to multiply this vector by any infinitesimal piece on the sides of this of this parallel pipette well I'll do it this way I will represent is as E of x, y, z E y x, y, z and E z x, y, z projections of this vector onto each particular point so any vector whatever is inside I can put it as a combination of three coordinate vectors why it's important for me for a very important thing because now I want to let's say calculate the flux which goes through this area I will use only the component of this vector E x plus E y plus E z I will use only E x because E y and E z are parallel to this vector because if I will multiply E times some kind of area s since E is equal to E x plus E y plus E z times s which is equal to E x times s plus E y times s plus E z times s but I know that E z is parallel to s now if this s is this particular side parallel and which means that projection onto a normal onto perpendicular to a service would be zero right so I don't have this same thing with y if it's along the y E y again it's parallel to this so if it's parallel there is no flux nothing goes through the surface and only E x actually remains because E s well first of all I assume that this particular parallel of the people is so small that my P is basically constant inside it so the vector here or vector here or inside or wherever it is vectors are the same obviously if delta x delta y and delta z are small enough I can assume that the density is exactly the same within this small parallel so E is the same and doesn't depend right now on whether it's inside and left and right or whatever it's exactly the same everywhere and if it's the same everywhere I can calculate flux through this particular area by basically multiplying E of x plus delta x yz I'm here right I'm talking about this particular thing so the flux goes here and this value has coordinates x plus delta x but y and z are the same so that's why I put this thing I will multiply it by this area by delta y times delta z the area is equal to delta y times delta z and this is the flux which goes through this right let's say right side so it's a right, left, top, bottom front and back six sides I will talk about each one separately so this is my flux through the right side now what is my flux to the left side well the value of intensity vector would be E of x, y, z right this is x plus delta x and this is just x on the left side I'm talking about left side what's important is these two vectors are almost the same they differ only a little bit but the normal to a surface is completely opposite here normal goes this way I probably didn't mention it normal should go outside of the surface so this is the closed surface normal goes here to the right and on the left one it goes to the left opposite and whenever I'm multiplying using this particular formula this vector is this way but normal is that way so if I'm talking about projection it will be the same by magnitude but it will be with a minus sign and the area would be exactly the same the left area is exactly the same so this is the area but the flux should be using the amount of intensity with a minus sign because the area is oriented opposite to this one they're two opposites so if I'm talking about flux obviously this goes this way and this goes that way from the central it goes outside that's why they're going to different direction that's why we have to subtract them to know what's exactly the total flux so the total flux in two surfaces right and left would be difference between these two guys which is equal to let me divide and multiply by delta x it would be x plus delta x yz minus e of x y z divided by delta x and multiplied by delta x delta y and delta z so I'm just multiplying divided by delta x with this difference and combine together this intensity now what is this I mean to those who are very much familiar with calculus and I presume that all my listeners are familiar with calculus whenever the delta x goes to zero we're talking about this thing infinitesimally small right this thing if you fix y and z that would be a derivative by x so it's called partial derivative because y and z so the total result this would be derivative of e x y z by dx times dx times dy times dz well I actually should put z differential then rather than delta so we're talking about all the delta coming down to zero so this is actually whenever this parallel people are small enough this is what I have from right and left flux okay now what will I have front and back same thing I will have de of x y z dy exactly the same thing so instead of x plus delta x I would have y plus delta y minus minus y and it would be the same thing so that would be there and finally the same thing for z by dz great so for every parallel people I will have this as a total flux which goes through this parallel people and finally I have to I have to summarize it right now you remember that my that my initial talk about sphere I was talking about the total flux through a sphere is equal to the charge inside flux was equal to charge inside divided by epsilon right okay now and I was talking that it doesn't really depend on the shape whether it's a sphere or a cube or whatever it's a parallel people so some of this the total flux should be equal to this and this is in this particular case if this is a value the volume is equal by the way delta x times delta y times delta z or dx dy dz and I have a density rule that what I would like to say is that the total result of their sound would be equal to total charge and total charge is equal to raw times dx dy dz divided by epsilon right now obviously dx dy dz will cancel out here and I have a very important formula almost finished with this important formula is d e of x y z put dy sorry put dx first plus d e partial derivative by y plus partial derivative dz is equal to raw divided by epsilon nice nice I will make it even nice now by the way if you are not familiar with partial derivatives and actually it's practically the same thing as plane derivatives with only one argument of multiple argument functions so this is function of three arguments but I'm differentiating only by one basically keeping y and z constant that's why it becomes basically partial derivative now but smart mathematicians or physicists actually decided to do well they have decided to basically abuse some notation they have said okay let's introduce a vector a pseudo vector it's not a vector actually nabla which is d by dx d by dy d by dz now these are not vectors but they look like a vector I mean it's basically a combination of three operators written as a vector now this is just notation so nabla is not a vector and it's not an operator it's just three operators together combined by a couple of commas and curly brackets so it's not a big deal but we can treat it as a vector whenever we do some operations for example what is nabla scalar product with vector e x, y, z now vector x, y, z is basically a combination of e, x x, y, z e, y x, y, z e, z x, y, z so it's three components and the scalar product is basically a multiplying by individual components and sum them together so this is exactly equal to d by dx so x, y, z now I can put x here but it doesn't really matter because we are talking about differentiating only by x so it would be the same plus d by dy e plus d by dz e now using this particular notation you can put nabla times e equals rho divided by x now this is a short notation for this now it's nothing but changing of the notation there is nothing more than that but it looks kind of nice so this is kind of a scalar product of a pseudo vector and the vector okay so basically that's it and this is something which is basically the form of the first Maxwell equation and I was trying to basically explain it again I did not prove it like all the different forms, shapes, etc however I think you have a feeling of what exactly the flux is and what's important is the total flux through any closed surface is basically can be on differential level on a very very small surface is equal to density of electric charge inside divided by epsilon well obviously something like a point charge if you take well what's the density if you have a single point charge in a sphere well in this particular point where the charge is it would be infinite so it's an abstraction but charges are not really charges are always spread around and we can always talk about density of the charge means charge divided by volume within which this charge is located okay in any case that's not something which you can learn in like university where you have a very rigorous presentation of this this is just to give you a feel of what the first Maxwell equation is now read the lecture and good luck thank you