Induction by simultaneous temporal change of b and motion of the conductor (Индукция одновременным изменением во времени b и движением проводника)

Let's now try to unite them together to look what will happen if simultaneously the configuration of the electromagnetic field changed in time and the magnetic field itself also changed in time. In such a case, both effects should be taken into account, both of them will induce parts of electric field intensity. If now we shall consider again the integral of Edl product around the closed loop, then we can split this relation into two parts. The first of them is what is going on with the frame, contour if only magnetic field changed in time. Another part of the Faraday's Law is what is going on with the contour if magnetic field is stable, static but the contour changes it's configuration. And finally we shall get this relation:

This relation describes, corresponds to the Faraday's Law in the case of moving conductor and the same relation is valid in the case when both coil configuration and magnetic flux density depends on time. In both cases this full relation is correct and should be used.

Let's look at some properties, general properties of the electromagnetic field when this universal (both flux density and coil configuration depend on time) case is consider. Using the vector magnetic potential:

(The Gauss theorem for curl of vector)

Or

The last expression is true for any surface so:

Conclusion: Curl of this parenthesis equal to zero. Why it's important? We have talked already about the vectors which have zero curl. So if the curl of function equal to zero, then we can express it as gradient of some function. Let's do it here also.

Introducing a suitable scalar function , we may write: that is a general expression which may be used in the case of static field, dynamic field, fields which changes in time, static field with the moving conductor.

H ere we can express separately vector E: the field intensity E here is the one which an observer would "see" when moving with the conductor.

An observer at rest then sees the field:

Unipolar generator (Униполярный генератор)

Here we have a rotating wheel, conducting wheel and external magnetic flux density, which is parallel to the axis of this wheel. This wheel is rotating with some angular velocity and there are brush contacts. One of this brush contacts is attached to the axis, another one is attached to the circular part of the wheel. This contour is vertical, it is not crossed by the magnetic field at all.

The result of this closed circuit is an induced EMF.

How the voltage may be induced in this contour? We have no changing magnetic flux. But the experiment show us there is the voltage.

The flux passing through the contour is constant! Ф=0

Velocity is a function of the distance to the center:

Electric field intensity:

So, the field really will be induced, but what to do with the flux? It's always equal to zero, theory tells us that in the universal case the induced electromotive force is equal to full derivative dФ/dt with the minus sign, not applicable for this case.

T he second impression we should look at this system even in more details. The contour is more complex that we have seen in the beginning. The current goes along this line, but then really it can't come through this line, because this is the moving object. And current comes really along the radius, but radius moves by itself. Finally, current will move along such curve line. Nevertheless, let's try to analyze the magnetic flux which is crosses this triangle. The area of this triangle is

T he magnetic flux in this case is:

The induced EMF is:

However, this explanation is not very persuasive. What is always true: Using the flux in this certain situation makes no sense.