We derive a general form of the induced electromotive force due to a time-varying magnetic field. It is shown that the integral form of Faraday's law of induction is more conveniently written in the covering space. Thus the differential form is shown to relate the induced electric field in the n<sup>th</sup> winding number to the (n+1)<sup>th</sup> time-derivative of the magnetic field.
Faraday’s law of induction in its differential and integral forms is a well-known standard topic which is discussed in many textbooks on electricity and magnetism [1-4]. Its integral form relates the closed line integral of the induced electric field to the negative time-derivative of the enclosed magnetic flux. This induced electric field creates an induced electromotive force which gives a magnetic field that, by Lenz’s law, opposes the change in the magnetic flux. Therefore the magnetic field is modified each time the path is traversed. A consequence of this is that the induced electromotive force (ε) is a sum over all contributions of multiple paths. The role of multiple paths in determining the final outcome was suggested long time ago by Feynman in his path integral method [
Consider an external time-varying magnetic field that passes through a circuit of resistance R. We derive the induced electromotive force (emf = ε) by the method of successive approximation. First, we pretend that the magnetic field is, and the emf is given by the negative rate of change of the magnetic flux as
and thus the induced current is
This induced current generates a magnetic field of its own, , whose direction is such that, by Lenz’s law, to oppose the change in the magnetic flux. Biot-Savart law ensures that this magnetic field can be written as
where is a vector whose magnitude depends on the geometry of the circuit. In the second step of the approximation, the total magnetic field through the circuit is
and thus, using Equation (1), the modified emf at the end of this step is
The above emf gives rise to a modified current given by
which in turn generates its own magnetic field, given by
where the prime denotes time derivative.
In the third step of the approximation, the total magnetic field that passes through the circuit is
which corresponds to another modified emf given by
In a similar manor, it is easy to show that the modified emf at the end of the fourth step of approximation is
where is the time-derivative of the magnetic field. Therefore, carrying the above steps further, one finds that the general formula for the emf is
It is instructive to write Equation (11) in terms of the self-inductance, of the circuit which is defined as
so the result is
The above equation shows that the induced electromotive force is written as a power series of self-inductance of the circuit and surface integrals over higher time-derivatives of the external magnetic field,. One may easily check that the term in the sum in Equation (13) has the expected unit, volt. Furthermore, the first term in the sum () gives the induced emf that is found in most standard electromagnetic textbooks [
The term of Equation (13) contributes to the total induced emf a quantity given by
and, as is well-known [
This may be viewed as a closed line integral around the loop (winding number) in the covering space whose polar angle. The necessity for a covering space was emphasized long time ago by AL-Jaber and Henneberger [
which gives the loop (winding number n) contribution to the induced electric field due to the time-derivative of the magnetic field. In the physical space, where the polar angle, one has to add all contributions coming from different loops (winding numbers) to get the induced electric field, namely
The above equation gives the integral form of Faraday’s law. It must be noticed that the closed line integral on is carried out on the physical space, while the closed line integral on the right is carried out on the covering space. The differential form of Faraday’s law is readily obtained by applying Stoke’s theorem to convert a closed line integral into a surface integral. In the covering space, one immediately gets
while in the physical space, the result is
The case n = 0 is readily obtained from the above two equations with the result
which is a well-known result.
In this paper, we derived a general form of the induced electromotive force using the method of successive approximation. It was shown that this induced electromotive force is a power series of the self-inductance of the circuit and of surface integrals of higher time-derivatives of the external magnetic field. The first term in the series is the familiar induced electromotive force and the higher order terms are contributions coming from different winding numbers in the covering space. The term in the series is accounted for by the closed line integral of an induced electric field around the loop (winding number) in the covering space. Therefore, the integral form of Faraday’s law was written in the covering space and the contribution coming from the winding number is proportional to the power of self-inductance and to the surface integral of the timederivative of the external magnetic field. In addition, the differential form of Faraday’s law relates the curl of the induced electric field coming from the winding number to the time-derivative of the external magnetic field. While in the physical space, the curl of the total induced electric field is related to the sum of all contributions coming from different winding numbers.