Lenz’s Law (правило Ленца)

Lenz’s Law climes that the electric field induced in a circuit due to a change in a magnetic field is directed to oppose the change of the flux.

Step by step we shall consider different cases, when this Faraday's Law forms special rules, which can be used to fulfill the real calculations of the electric field. Let’s start with the case when the electric field is induced by temporal change of B. It means that electromagnetic system is stable, doesn't move and only B (the flux density) depends on time.

Induction by a temporal change of b (Индукция за счёт временного изменения b)

First of all, let’s look at some very important definitions of the electromagnetic field theory. The Maxwell equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field:

It is partial derivative, not full derivative. That is important just for this consideration, for this part of our theory. The initial equation in the Maxwell theory just operates with this partial derivative.

The magnetic flux is defined:

There is an illustration, what is it the flux: magnetic flux density lines here crosses the area which is limited by some contour. All these lines forms the magnetic flux. So, let’s try to transform this Maxwell equation. For this purpose let’s integrate the scalar product. I’m looking at this contour, to the small elements (на картинке) of the contour dl and probably, it’s not clear does it exist or not. But probably, the electric field intensity also exists in this space. Let’s try to calculate this integral . By the way, this integral will be just a voltage, which is applied to the contour (this definition was discussed in our first lecture, where we link together the scalar electric potential and electric field intensity).

That is special case of the Gauss theorem, it’s applied just for our case: . That is Faraday’s Law in differential form.

Or we can replace this integral with voltage, so: it’s electromotive force (EMF-ЭДС) or voltage (в нашем учебнике это напряжение заменено на понятие ЭДС). That is Faraday’s Law in integral form.

(Честно, не совсем понял, почему где интеграл это дифференциальная форма, а где производная это интегральная форма. Наверно всё связано с ротором и что интеграл Edl=U.)

Induction through the motion of the conductor (Индукция за счёт движения проводника)

We assumed from the beginning that B doesn't dependent on time, it's constant. It's distributed somehow over the space, so it may have different values in different points, but it doesn't dependent on time.

C oming back to the previous expressions, it will mean: or will be exactly equal to zero, because we have seen only partial derivative. Partial derivative analyses only the time dependent functions. So, if it is constant, then it will give us zero. Nevertheless, sometimes even in static fields the electromotive force may be induced. To understand the main principles of such induction let's start with this expression:

It's definition of the magnetic flux density vector B. It is introduced in this theory just in this way, just using this expression. In the left side we see the force, which is applied to the particle which moves in the external magnetic field (Q - is a charge of this particle). We can measure all these experimentally, measure charge, velocity and force. If we know all 3 values, then we can say in this point of the space the flux density is equal to some value and this value may be found from this expression.

Let's look at another relation which also defines a force, which may be applied to the charged particle . In this expression doesn't matter does it move particle or it has the same position always, which doesn't dependent on time. That is a definition of electric field intensity.

This effect may be explained by assuming: . We can twit this product in brackets here as a electric field intensity. It appears, that if some system moves in the magnetic field, the electric field intensity is generated in this moving system. And that is a simple explanation which may be used to find consequences, which links together the field intensity and the magnetic field flux density.

W e are going to find the electromotive force which may be generated in some contour. Let's substitute the vector product in this relation instead of electric field intensity. Moving a thin conducting contour in static magnetic field we shall get:

F or this purpose let's remember from mathematics. What does this expression physically meaning? Let's look at the physical meaning of this product is shown in the picture. The red line corresponds to a product. Why dt came here? I want to plot some geometrical figure in this space, I will use one side to be equal to dl, but another side also should have damages of meters, units of distant. For this purpose i added to velocity vector dt. Here we see vector product, if these two vectors are normal to each other, then there will be a rectangle and the product dl and vdt will be area of this rectangle. If they are not normal to each other, there is an angle between them. Nevertheless, there will be a parallelogram and the vector product of dl and this vdt will be the area of this parallelogram. Rectangle is easier to understand and doesn't change the result. So, this rectangle has an area which vector dl covers during its movement. This product is really an element of area which is covered by moving conductor. It is expressed here:

Electromotive force now may be transformed to a new relation. dt now is moved to outside the integral, because in general case B (the flux density) also may dependent on time. In such a case this transformation will be impossible, but just at this stage of our consideration we talk about induction through the motion of the conductor, only motion of the conductor in the magnetic field, which doesn't dependent on time. So, time derivative doesn't act on the flux density.The EMF induced in the contour:

But now the flux here is different sheathed (заключённый) not by the partial, but by the full derivative (d/dt).

If the magnetic field doesn't depend on time, the partial derivative will gives here simply zero. The magnetic field distribution doesn't depend on time!

W e can conclude that electromotive force may be found by integrating the electric field which is presented as . This property of electromagnetic field may be used for calculations, sometimes it's easier to consider the total flux, which is coupled with the contour, sometimes it's more convenient to calculate the electric field intensity as some intermediate variable and then its possible to integrate this E vector around the closed loop.

The special case is a conductor, moving in the external magnetic field. There is two properties of this magnetic system which has controversial properties. The first of them: the conductor is not a closed loop, so the current can't flow in principle, current may flow only in closed circuit. On the other hand, just now we have found that, if this conductor moves in the external magnetic field then the electric field inside will be induced. We know that conductor is described by the Ohm's Law, the current density is always proportional to the electric field intensity and the conductivity is the coefficient of proportionality. So, it looks like the current should flow. Otherwise, there is some mistake in our consideration. Of course, the first decision that there is no current is correct. But how to treat this expression, that electric field should induced. Let's look at the same problem in developing, how it's developed from the beginning. Let's assume that this conductor doesn't move and then we started to moved with the velocity of v. Initially, when it was stable of course, the velocity was equal to zero and the electric field wasn't induced and so the current should not flow through the conductor. And there is no any contradictions in such a case. When we started to move this conductor in the first moment, the electric field intensity was induced and the current started to flow in the direction of the electric field. But there will be a consequence of this current. Current flow means, let's assume that is a metal, so the electrons started to move along the conductor from beginning of the conductor to the end. So the charge difference are became inside the conductor. At one top of the conductor the electrons are concetrated now, at another top of this conductor the nuclear which have positive charge are not compensated by the electron. That is why this conductor became as if it is a charged capacitor and these separated charges generates their own electric field, we should have a super position of two electric fields. One of them is induced by moving conductor. Second is field which is induced by separated charges. These two fields will be exactly equal to each other. If the velocity is constant, then there will be no any current.

I f the last 3 vectors are normal to each others than:

Let's consider another electromagnetic system which is closed. We see a frame which consists of four conductors consists of four sides, current is possible. Let's look what will happen if we shall start to move this frame in the external magnetic field.

We shall set a distribution of the magnetic field in this space. Let's consider the magnetic field is uniform. What will happen in such a case? We're moving this frame, the electric field which is induces at the front and back are normal to the lengths of the side, that's why this electric field doesn't induce any voltage, electromotive force.

Now look at the left and right side, the electric field which is induced right side is identical, has the same direction, value, because we assume that the magnetic field is the uniform. If the contour moves in the uniform magnetic field no EMF is induced because of mutual canceling of partial electromotive forces:

Another case:

three sides of the contour are static, doesn't changed in time and only one side changes its position in time. We shall assume that this side moves with the constant velocity, the direction of the movement is normal to the direction of the magnetic field flux density. In such a case, the voltage will be induced. If we shall remember about the relations between field intensity and flux density, we can immediately calculate the induced voltage.

B ut also we can use alternative way of voltage (EMF) definition. This force may be found as

Sign minus need only to incorporate Lenz's rule in the theory. Positive voltage or negative? It depends on how should we attach our voltmeter to the considered system.

During this important process of energy transformation, mechanical energy is transformed into electrical energy.

T here is another situation when the EMF may be generated in the rectangular frame which has the same cross section, area and configuration. The example here is just a rotation of the rectangular frame in the external uniform magnetic field. In such a case, the flux which is coupled with this rectangular frame may be described as a sinusoidal or cosinusoidal function, which depends also on the omega (angular frequency of rotation).

This is the principle of alternating current generators.