 Engines for electric cars, magnetic resonance imaging in medicine, electric candles in the kitchen, the charger of your smartphone, radio, Wi-Fi and so on. Any device that exploits electricity or magnetism is fundamentally based on the Maxwell equations. The goal of this video is not to derive the Maxwell equations theoretically or experimentally but to present them as simple and understandable as possible. I will also briefly explain the mathematics that occur in the Maxwell equations. However you should know what a partial derivative is and what an integral is before you continue to watch the video. To understand all Maxwell equations you first need to know what an electric and a magnetic field is. Consider a large electrically charged sphere with a charge capital Q and a small sphere with a charge small q. The electric force between these two spheres, which are the distance r from each other, is given by Coulomb's law. Here 1 over 4 pi epsilon 0 is a constant pre-factor with the electric field constant epsilon 0, which provides the right unit of force, the unit Newton. Now what if you know the value of the big charge and want to know the value of the force that the big charge exerts on the small charge? However you do not know the exact value of this little charge or you intentionally leave that value open and only want to look at the electric force exerted by the big charge. So you have to somehow eliminate the small charge q from Coulomb's law. To do so you simply divide Coulomb's law by q on both sides. This way on the right hand side the small charge drops out and instead lands on the left hand side of the equation. The quotient on the left hand side f over q is defined as the electric field E of the source charge q. By calling it source charge we imply that the charge q is the source of the electric field. The electric field E thus indicates the electric force which would act on a small charge when placed at the distance r to the source charge. So far only the magnitude that is the strength of the electric field has been considered without taking into account the exact direction of the electric field. However the Maxwell equations are general and also include the direction of the electric field. So we have to turn the electric field into a vector. Vectors are shown in boldface. Handwritten mostly with a little arrow above the letter to distinguish it from scalar, poor numbers. I omit the arrows because they make the equations unnecessarily ugly. The electric field E as a vector in three dimensional space has three components EX, EY and EZ. Let's look at the first component. The first component depends on the space coordinates X, Y, Z and it is the magnitude of the electric field in X direction. That is depending on which concrete location is used for X, Y, Z the value EX is different. The same applies to the other two components EY which indicates the magnitude in the Y direction and EZ which indicates the magnitude in the Z direction. The components of the electric field thus indicate which electric force would act on a test charge at a specific location in the first, second or third spatial direction. Another important physical quantity found in the Maxwell equations is the magnetic field B. Experimentally it is found that a particle with electric charge Q moving at the velocity V in an external magnetic field experiences a magnetic force which deflects the particle. The force on the particle increases in proportion to its charge Q and its velocity V that is doubling the charge or the speed doubles the force on the particle. But not only that the force also increases in proportion to the applied magnetic field. The unit of this quantity must be such that the right hand side of the equation gives the unit of force that is Newton or kilogram meter per second squared. By a simple transformation you will find that the unit must be kilogram per ampere second squared. This is what we call the unit of Tesla and we call B magnetic flux density or short magnetic field. The magnetic flux density describes the external magnetic field and thus determines the magnitude of the force on a charged particle. The equation Q times V times B for the magnetic force on a charged particle represents only the magnitude of the force. In order to formulate the magnetic force with vectors analogous to the electric force, the force, the velocity and the magnetic field are expressed in vector form. Now the three quantities are not scalars but three-dimensional vectors with components in the x, y and z direction. Now the question is how should the velocity vector V be vectorially multiplied by the magnetic field vector B? If you look more closely at the deflection of the charge in the magnetic field you will notice that the magnetic force always points in a direction orthogonal to the velocity and to the magnetic field lines. This orthogonality can be easily taken into account with a so-called cross product. The cross product of two vectors V and B is defined so that the result of the cross product V cross B which is a vector is always orthogonal to the two vectors V and B. The first component is VIBZ minus VZBI. The second component is VZBX minus VXBZ and the third component of the cross product is VXBY minus VYBX. For the force to be always orthogonal to V and B we take the cross product of V and B in our equation. So in vector form the magnetic force is generally given by Q times V cross B. The charge is an ordinary scalar factor. As you can see the physical quantity B describes the strength of the magnetic field which leads to a deflection of moving charges. Now you have learned the two important physical ingredients found in the Maxwell equations. The electric field E and the magnetic field B. Both are so-called vector fields. This means that to each location XYZ in space you can assign an electric and a magnetic field vector indicating both magnitude and direction of the electric and magnetic fields. There are a total of four Maxwell equations. These four Maxwell equations can be represented in two different ways. There is the so-called integral form which expresses the Maxwell equations with integrals and the differential form which expresses the Maxwell equations with derivatives. While the differential form of Maxwell's equations is useful for calculating the magnetic and electric fields at a single point in space the integral form is there to compute the fields over an entire region in space. The integral form is well suited for the calculation of symmetric problems such as the calculation of the electric field of a charged sphere, a charged cylinder or a charged plane. The differential form is more suitable for the calculation of complicated numerical problems using computers or for example for the derivation of the electromagnetic waves. In addition the differential form looks much more compact than the integral form. Both forms are useful and can be transformed into each other using two mathematical theorems. One theorem is called divergence integral theorem and the other one curl integral theorem. If you understand the two theorems it will be easier for you to understand the Maxwell equations. Let's first look at the divergence integral theorem. This is what the divergence integral theorem looks like in its full splendor. First let's look at the right-hand side of the equation. The A represents a surface enclosing any volume for example the surface of a cube, a sphere or the surface of any three-dimensional body you can think of. The small circle around the integral indicates that this surface must satisfy a condition. The surface must be closed. In other words it must not contain any holes so that the quality is met mathematically. The surface A is thus a closed surface. The F is a vector field and represents either the electric field or the magnetic field when considering the Maxwell equations. So it is a vector with three components. DA is an infinitesimal surface element that is an infinitely small surface element of the considered surface A. As you may have already noticed the A in the DA element is shown in boldface so it is a vector with a magnitude and a direction. The magnitude of the DA element indicates the area of the small piece of the surface. The DA element is orthogonal to the surface and by definition points out of the surface. The dot between the vector field and the DA element represents the so-called scalar product. The scalar product is a way to multiply two vectors. So here the scalar product between the vector field and the DA element is formed. The scalar product is defined as follows. As you can see from the definition the first, second and third components of the two vectors are multiplied and then added up. The result of the scalar product is no longer a vector but an ordinary number as so-called scalar. To understand what this number means you must first know that any vector can be written as the sum of two other vectors. One vector that is parallel to the DA element let's call it F parallel and another vector that is orthogonal to the DA element let's call it F orthogonal. Another mathematical fact is that the scalar product of two orthogonal vectors always yields zero which means that in our case scalar product between the part F orthogonal and the DA element is zero. However the scalar product between the part F parallel and the DA element is generally not zero. So now you can see what the scalar product on the right hand side of the equation does. It just picks out the part of the vector field that is exactly parallel to the DA element. The remaining part of the vector field that points in the orthogonal direction is eliminated by the scalar product. Subsequently the scalar products for all locations of the considered surface A are added up. That is the task of the integral. The right hand side of the divergence integral theorem thus sums up all the components of the vector field F that flow into or flow out of the surface A. Such an integral in which small pieces of a surface are summed up is called surface integral. If as in this case the integrand is a vector field this surface integral is called the flux phi of the vector field F through the surface A. This description is based on what this surface integral means. It measures how much of the vector field F flows out or flows into a considered surface A. If the vector field F and the surface integral is an electric field E then this surface integral is called electric flux through the surface A. And if the vector field F is a magnetic field B the surface integral is called magnetic flux through the surface A. Now let's look at the left hand side of the theorem. V is a volume but not any volume it is the volume enclosed by the surface A. dv is an infinitesimal volume element in other words an infinitely small volume piece of the considered volume V. The upside down triangle is called nabla operator and it has three components like a vector. Its components however are not numbers but derivatives corresponding to the space coordinates. The first component is the derivative with respect to x the second component is the derivative with respect to y and the third component is the derivative with respect to z. Such an operator like the nabla operator only takes effect when applied to a field and that also happens in this integral. The nabla operator is applied to the vector field by taking the scalar product between the nabla operator and the vector field. As you can see it is the sum of the derivatives of the vector field with respect to the space coordinates x, y and z. Such a scalar product between the nabla operator and a vector field F is called the divergence of the vector field F. The result at the location x, y, z is no longer a vector but a scalar which can be either positive negative or zero. If the divergence at location x, y, z is positive then there is a source of a vector field F at this location. If this location is enclosed by a surface then the flux through the surface is also positive. The vector field so to speak flows out of the surface. If the divergence at location x, y, z is negative then there is a sink of a vector field F at this location. If this location is enclosed by a surface then the flux through the surface is also negative. The vector field flows into the surface. If the divergence at location x, y, z disappears then that location is neither a sink nor a source of the vector field. The vector field does not flow out or into or it flows in as much as out so the two amounts cancel each other out. Subsequently the divergence that is the sources and sinks of the vector field is summed up at each location within the volume using the integral. Such an integral where small pieces of volume are summed up is called volume integral. So let's summarize the statement of the divergence integral theorem. On the left side is the sum of the sources and sinks of the vector field within a volume and on the right side is the total flux of the vector field through the surface of that volume and the two side should be the same. The divergence integral theorem thus states that the sum of the sources and sinks of a vector field within a volume is the same as the flux of the vector field through the surface of that volume. Now consider the second important theorem necessary for understanding the Maxwell equations the curl integral theorem. If you understand the divergence integral theorem then the curl integral theorem should not be totally cryptic to you. You already know the vector field F the scalar product the nabla operator and the DA element should also be familiar to you now. First let's look at the right hand side of the equation the L is a line in space. The circle around the integral sign indicates that this line must be closed that is it should form a loop whose beginning and end are connected. The DL is an infinitesimal line element of the loop so an infinitely small piece of the line. Again you should notice that the DL element is shown in boldface. It's a vector with a magnitude and a direction. The magnitude of the DL element indicates the length of this small line and its direction points along the line L. Now the scalar product between the vector field F and the line element DL is formed. You already know what the task of the scalar product is. First split up the vector field into two parts into F parallel which is parallel to the DL element and into F orthogonal which is orthogonal to the DL element. The scalar product with a DL element eliminates the orthogonal component without touching the part of the vector field parallel to the DL element. Since at each location the DL element points along the line only the part of the vector field that runs along the line L is considered in the scalar product. The other part of the vector field drops out. Then the scalar products for each location on the loop are summed up using the integral. Such an integral in which small line elements are summed up is called line integral. Now you know what happens on the right hand side of the curl integral theorem. The line integral measures how much of the vector field F runs along the line L. Because the line is closed this scalar product returns to the same point where the summation started. The closed line integral thus indicates how much of the vector field F rotates along the loop L. If the vector field F in this line integral is an electric field E then this line integral is referred to as electric voltage along the line L. On the other hand when the vector field F is a magnetic field B the line integral is called magnetic voltage along the line L. The voltage in the case of an electric field is proportional to the energy that a positively charged particle gains as it passes the line L. A negatively charged particle on the other hand loses this energy as it passes the line L. The line integral of the electric field that is the voltage measures the energy gain or energy loss of charged particles as they pass through the line L under consideration. Now you should have understood the right hand side of the curl integral theorem. Let's look at the left hand side now. Here the surface A occurs again. This surface unlike the divergence integral theorem must not be a closed surface but it is simply the surface enclosed by the line L. The A is again an infinitesimal piece of the surface A and at any location it is orthogonal on that surface. In addition here comes the cross product which you have already met when we discussed the magnetic force. Here the cross product is formed between the nubla operator and the vector field F. In addition to the scalar product it is the second way to multiply two vectors. This cross product between the nubla operator and the vector field F is called the curl of the vector field F. The result in contrast to the scalar product is again a vector field. This new vector states how much of the field F rotates around a point within the surface A. Then the scalar product is formed between the new vector field nubla cross F with the infinitesimal surface element DA. Thus as you already know only the part of nubla cross F is picked out which runs parallel to the surface element. Since the surface element is orthogonal to the surface A the scalar product picks out only the part of the vector field nubla cross F which is also orthogonal on the surface A. Subsequently all scalar products within the surface A are summed up by means of the integral. Let's summarize the statement of the curl integral theorem. On the right hand side the vector field F is summed up along a line L. Thus the rotation of the vector field around the enclosed surface is considered. On the left hand side the curl of the vector field F is summed up at each individual point within the surface. Both sides should be equal according to the theorem. The curl integral theorem thus states that the total curl of a vector field F within the surface A corresponds to the rotation of the vector field F along the edge L of that surface. Well it is somehow clear that the rotation of the vector field inside of the surface cancels in the summation in only the rotation of the vector field along the edge L remains. With the acquired knowledge you are now ready to fully understand the Maxwell equations. Here we go. This is the first Maxwell equation in integral form. The left hand side of the Maxwell equation should be familiar to you. It is a surface integral in which the electric field E occurs. This integral measures how much of the electric field comes out of or enters the surface A. The integral thus represents the electric flux through the surface A. On the right hand side is the total charge Q which is enclosed by the surface A divided by the electric field constant which provides the correct unit. The first Maxwell equation states that the electric flux phi through a closed surface A corresponds to the electric charge Q enclosed by the surface. By the way, Coulomb's law is a special case of the first Maxwell equation. With the previously learned divergence integral theorem, which combines a volume integral with the surface integral, the surface integral on the left hand side of the first Maxwell equation can be replaced by a volume integral of the divergence of the electric field. The enclosed charge Q can also be expressed with a volume integral. By definition, charge density rho is charge per volume. Bring the volume to the other side. The volume V can generally be written in the form of a volume integral. That is, the volume integral of the charge density rho over volume V is the charge enclosed in that volume. This turns the right hand side of the Maxwell equation into a volume integral. As you can see, we integrate over the same volume V on both sides. For this equation to be satisfied for an arbitrarily chosen volume V, the integrants on both sides must be equal, the right integrant being multiplied by the constant 1 over epsilon 0. And now you have discovered the differential form of the first Maxwell equation. The divergence of E is equal to rho over epsilon 0. On the left hand side of the differential form, you can see the divergence of the electric field. You know that at a specific point in space, it can be positive, negative or zero. The sign of the divergence determines the type of the charge at the considered point in space. If the divergence is positive, then the charge density rho at this point in space is positive and thus also the charge. In this point of space, there is therefore a positive charge, which is a source of the electric field. If the divergence is negative, then the charge density rho is negative and thus also the charge. At this point of space, there is therefore a negative charge, which is a sink of the electric field. If the divergence is zero, then the charge density rho is zero as well. At this point in space, there is either no charge or there is just as much positive charge as negative, so the total charge at this point is cancelled out. At this point in space, an ideal electric dipole could be located. The first Maxwell equation in differential form states that the electric charges are the sources and sinks of the electric field. Charges generate the electric field. This is the second Maxwell equation in integral form. There is nothing unfamiliar in this equation. Everything should look familiar to you now. On the left hand side, you see a surface integral over A. Now, not an integral of an electric field as in the first Maxwell equation, but an integral of a magnetic field B. According to the equation, the magnetic flux through the closed surface A is always zero. The second Maxwell equation states that there are always just as many magnetic field vectors coming out of a surface as there are vectors entering the surface. With the divergence integral theorem, the surface integral can be transformed into a volume integral. This way, the divergence of the magnetic field comes into play. This integral shall be zero. The integral for any volume V is only always zero if the integral is zero. In this way, the second Maxwell equation emerges in its differential form. Divergence of B is equal to zero. If the divergence is zero, this means that at each point in space, x, y, z, there is either no magnetic charge, also called a magnetic monopole, or there is just as many positive magnetic charge as negative, so the total charge at that point cancels out, such as an ideal magnetic dipole, which always has both a north and a south pole. The north pole corresponds to a positive magnetic charge, and the south pole corresponds to a negative magnetic charge. Since there are no magnetic monopoles, there are no separated sources and sinks of the magnetic field. The second Maxwell equation in differential form states that there are no magnetic monopoles that generate a magnetic field. Only magnetic dipoles can exist. The second Maxwell equation, like the other Maxwell equations, is an experimental result. That is, if someday a magnetic charge should be found, for example a single north pole without a corresponding south pole, then the second Maxwell equation would have to be modified. Then the Maxwell equations would look even more symmetrical, more beautiful. This is what the third Maxwell equation looks like in an integral form. You probably already know the third Maxwell equation under the name of Faraday's law of induction. This right here is the most general form of the law of induction. On the left-hand side is a line integral of the electric field E over a close line L, which borders the surface A. This integral sums up all the parts of the electric field that run along the line L, which means it sums up how much of the electric field rotates along the line. The integral corresponds to the electric voltage U along the line L. On the right side there is a surface integral of the magnetic field B over an arbitrary surface A. This integral corresponds to the magnetic flux phi through the surface A. This magnetic flux is differentiated with respect to time t. How to differentiate a function as, in this case, the magnetic flux with respect to time, you should have a down pad. The time derivative of the magnetic flux indicates how much the magnetic flux changes as time passes. So it's the temporal change of the magnetic flux. The larger the change of the magnetic flux, the greater the rotating electric field. The minus sign takes into account the direction of the rotation. If the change in the magnetic flux is positive, the electric voltage is negative. If the change in the magnetic flux is negative, the electric voltage is positive. The electric voltage and the change of the magnetic flux thus behave opposite to each other. The minus sign ensures energy conservation. Maybe you know that by the name Lenz law. As you can see, according to this Maxwell equation, rotating electric field produces time varying magnetic field and vice versa. The Lenz law now states that the magnetic flux, which is generated by the rotating electric field, counteracts its cause. Because if it was not the case, the rotating electric field would amplify itself and thus generate energy out of nowhere. That is impossible. So let's summarize. The third Maxwell equation states that the electric voltage along a closed line corresponds to the change in magnetic flux through the surface bordered by that line. In other words, a change in the magnetic flux through the surface A creates an electric voltage along the edge of A. Consider another important special case. If the magnetic field does not change in time, the right side of the Maxwell equation will be eliminated. Then the equation states that the electric voltage along a closed line is always zero. So there is no rotating electric field as long as the magnetic field doesn't change over time. If an electron passes the closed line L, it would not change its energy. Because as you learned, electric voltage indicates how much energy a charge gains or loses when it passes a line. In this case, the voltage is zero, therefore no change in energy. With a curl integral theorem, you can transform the integral form into the differential form. This theorem connects a line integral with a surface integral. To do this, simply replace the line integral with the surface integral. This brings the curl of E into play. On the other side, you may pull the time derivative inside the integral. Since the equation applies to any surface A, the integrants on both sides must be equal. And just now, you have discovered the differential form of the third Maxwell equation. Curl of E is equal to the negative time derivative of the magnetic field. The third Maxwell equation in differential form states that the changing magnetic field B causes rotating electric field E and vice versa in such a way that the energy conservation is fulfilled. Let's move on to the fourth, the last Maxwell equation. What kind of an integral is on the left side? Exactly, a line integral of the magnetic field B along the closed line L. This is the definition of the magnetic voltage U. On the right-hand side occurs the electric field constant epsilon 0 and the magnetic field constant mu 0. They ensure that the unit on both sides of the Maxwell equation is the same. In addition, something new occurs here, the electric current I. When electric charges flow along a conductor, they generate a current I. Also, there is another sum end. You know the surface integral of the electric field. This is the electric flux through the surface A. In addition, a time derivative is ahead of the electric flux, so the whole thing is the temporal change of the electric flux. On the right side, there are two summands, one contribution by the current and one contribution by the change of the electric flux. Thus, the fourth Maxwell equation states that the rotating magnetic field is generated first by the electric currents through the surface A and second by the changing electric field. Let us now derive the differential form. With the curl integral theorem, you can transform the line integral into a surface integral, thus bringing the curl of the magnetic field B into play. Now we have to express the current I with the surface integral so that we get a single integrand on the right. We can do that simply by using the current density J. It indicates the current per area through which the current flows. Consequently, the current can also be written as the surface integral of the current density J over the surface A. Note that here in the integral, the scalar product of the current density is taken with the DA element. So we pick only the part of the current density vector that runs parallel to the DA element. Only this parallel part of the current density contributes to the current through the surface A. You can pull the time derivative inside the integral. The two surface integrals can now be combined into one because we integrate over the same surface A. For the equation to be satisfied for any surface A, the integrants on both sides must be equal. And you already have the differential form of the fourth Maxwell equation that you are looking for. Curl of B is equal to mu zero times J plus mu zero epsilon zero times the change of the electric field over time. The differential formulas states that the curl of the magnetic field B at a point in space is caused in two ways. By the current density J and by an electric field changing at this point in space. Let us summarize the four Maxwell equations of electrodynamics in their compact form, in their differential form. The first Maxwell equation, divergence of E is equal to charge density over epsilon zero. Electric charges generate an electric field. The second Maxwell equation, divergence of B is equal to zero. There are no separate magnetic charges, thus no magnetic monopoles. The third Maxwell equation, curl of E is equal to negative temporal change of the magnetic field B. A changing magnetic field creates a rotating electric field and vice versa. The fourth Maxwell equation, curl of B is equal to mu zero times J plus mu zero epsilon zero times the change of the electric field over time. Electric currents and a changing electric field generate a magnetic field. So now you finally learned the foundation of all electrodynamics. Isn't it amazing how much knowledge and how many technical applications are contained in these four equations? By the way, Maxwell equation still hides something interesting that can be revealed by a few steps of transforming the equations. Electromagnetic waves, light. But that's a topic for another video. With this in mind, bye and see you next time.