According to Faraday’s law, in any closed linear path in space, when the magnetic flux φ surrounded by the path varies with time, a voltage is induced around the path equal to the negative rate of change of the flux in webers per second.
V = - ∂φ/ ∂t Eq. (2-2)
The minus sign denotes that the direction of the induced voltage is such as to produce a current opposing the flux. If the flux is changing at a constant rate, the voltage is numerically equal to the increase or decrease in webers in 1 s. The closed linear path (or circuit) is the boundary of a surface and is a geometric line having length but infinitesimal thickness and not having branches in parallel. It can vary in shape or position.
If a loop of wire of negligible cross section occupies the same place and has the same motion as the path just considered, the voltage will tend to drive a current of electricity around the wire, and this voltage can be measured by a galvanometer or voltmeter connected in the loop of wire. As with the path, the loop of wire is not to have branches in parallel; if it has, the problem of calculating the voltage shown by an instrument is more complicated and involves the resistances of the branches.
For accurate results, the simple Eq. (2-2) cannot be applied to metallic circuits having finite cross section. In some cases, the finite conductor can be considered as being divided into a large number of filaments connected in parallel, each having its own induced voltage and its own resistance.
In other cases, such as the common ones of D.C. generators and motors and homopolar generators, where there are sliding and moving contacts between conductors of finite cross section, the induced voltage between neighboring points is to be calculated for various parts of the conductors.
These can then be summed up or integrated. For methods of computing the induced voltage between two points, see text on electromagnetic theory.
In cases such as a D.C. machine or a homopolar generator, there may at all times be a conducting path for current to flow, and this may be called a circuit, but it is not a closed linear circuit without parallel branches and of infinitesimal cross section, and therefore, Eq. (2-2) does not strictly apply to such a circuit in its entirety, even though, approximately correct numerical results can sometimes be obtained.
If such a practical circuit or current path is made to enclose more magnetic flux by a process of connecting one parallel branch conductor in place of another, then such a change in enclosed flux does not correspond to a voltage according to Eq. (2-2).
Although it is possible in some cases to describe a loop of wire having infinitesimal cross section and sliding contacts for which Eq. (2-2) gives correct numerical results, the equation is not reliable, without qualification, for cases of finite cross section and sliding contacts. It is advisable not to use equations involving directly on complete circuits where there are sliding or moving contacts.