The Electric Generator

Figure 8.7.1 shows a rudimentary generator, which serves as to illustrate the relevant points. This generator consists of a planar loop that rotates around the

axis; therefore, the rotation can be parameterized in

. In this case, the direction of rotation is specified to be in the

direction. The frequency of rotation is

; that is, the time required for the loop to make one complete revolution is

. We assume a time-invariant and spatially-uniform magnetic flux density

where

and

are both constants. For illustration purposes, the loop is indicated to be circular. However, because the magnetic field is time-invariant and spatially-uniform, the specific shape of the loop is not important, as we shall see in a moment. Only the area of the loop will matter.

The induced potential is indicated as

with reference polarity as indicated in the figure. This potential is given by Faraday’s Law:

Here

is the magnetic flux associated with an open surface

bounded by the loop:

As usual,

can be any surface that intersects all magnetic field lines passing through the loop, and also as usual, the simplest choice is simply the planar area bounded by the loop. The differential surface element

is

, where

is determined by the reference polarity of

according to the “right hand rule” convention from Stokes’ Theorem. Making substitutions, we have

where

is the area of the loop.

To make headway, an expression for

is needed. The principal difficulty here is that

is rotating with the loop, so it is time-varying. To sort this out, first consider the situation at time

, which is the moment illustrated in Figure 8.7.1. Beginning at the “

” terminal, we point the thumb of our right hand in the direction that leads to the “

” terminal by traversing the loop;

is then the direction perpendicular to the plane of the loop in which the fingers of our right hand pass through the loop. We see that at

,

. A bit later, the loop will have rotated by one-fourth of a complete rotation, and at that time

. This occurs at

. Later still, the loop will have rotated by one-half of a compete rotation, and then

. This occurs at

. It is apparent that

as can be confirmed by checking the three special cases identified above. Now applying Faraday’s Law, we find

For notational convenience we make the following definition:

which is simply the time derivative of

divided by

so as to retain a unit vector. The reason for including a change of sign will become apparent in a moment. Applying this definition, we find

Note that this is essentially the definition of the radial basis vector

from the cylindrical coordinate system (which is why we applied the minus sign in Equation 8.7.6). Recognizing this, we write

and finally

If the purpose of this device is to generate power, then presumably we would choose the magnetic field to be in a direction that maximizes the maximum value of

. Therefore, power is optimized for

polarized entirely in some combination of

and

, and with

. Under that constraint, we see that

varies sinusoidally with frequency

and exhibits peak magnitude

It’s worth noting that the maximum voltage magnitude is achieved when the plane of the loop is parallel to

; i.e., when

so that

. Why is that? Because this is when

is most rapidly increasing or decreasing. Conversely, when the plane of the loop is perpendicular to

(i.e.,

)

is maximum but its time-derivative is zero, so

at this instant.

• “Electric generator” on Wikipedia.