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.
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