# 6.5 The Electromagnetic Generator

The electromagnetic generator is the converse of the electromagnetic motor. The principles of operation of both instruments may be explained with the help of Fig. 6-11. A permanent magnet is used to produce a static magnetic field B in the slot between its two poles. When a current is passed through the conducting loop, as depicted in Fig. 6-11(a), the current flows in opposite directions in segments 1–2 and 3–4 of the loop. The induced magnetic forces on the two segments are also opposite, resulting in a torque that causes the loop to rotate about its axis. Thus, in a motor, electrical energy supplied by a voltage source is converted into mechanical energy in the form of a rotating loop, which can be coupled to pulleys, gears, or other movable objects.

# Figure 6-11 Principles of the ac motor and the ac generator. In (a) the magnetic torque on the wires causes the loop to rotate and in (b) the rotating loop generates an emf.

(a) ac motor

Figure 6 11-a Full Alternative Text

(b) ac generator

Figure 6 11-b Full Alternative Text

If instead of passing a current through the loop to make it turn the loop is made to rotate by an external force, the movement of the loop in the magnetic field produces a motional emf, V e m f m, as shown in Fig. 6-11(b). Hence, the motor has become a generator, and mechanical energy is getting converted into electrical energy.

Let us examine the operation of the electromagnetic generator in more detail using the coordinate system shown in Fig. 6-12. The magnetic field is

B=z ˆ B 0,(6.30)

and the axis of rotation of the conducting loop is along the x axis. Segments 1–2 and 3–4 of the loop are of length l each, and both cross the magnetic flux lines as the loop rotates. The other two segments are each of width w, and neither crosses the B lines when the loop rotates. Hence, only segments 1–2 and 3–4 contribute to the generation of the motional emf, V e m f m.

# Figure 6-12 A loop rotating in a magnetic field induces an emf.

Figure 6 12 Full Alternative Text

As the loop rotates with an angular velocity ω about its own axis, segment 1–2 moves with velocity u given by

u=n ˆ ω w 2,(6.31)

where n ˆ, the surface normal to the loop, makes an angle α with the z axis. Hence,

n ˆ×z ˆ=x ˆ sin α.(6.32)

Segment 3–4 moves with velocity −u. Application of Eq. (6.26), consistent with our choice of n ˆ, gives

V emf m=V 14=∫2 1(u×B)⋅d l+∫4 3(u×B)⋅d l=∫−l/2 l/2[ (n^ω w 2)×z^B 0 ]⋅x^ d x+∫l/2−l/2[ (−n^ω w 2)×z^B 0 ]⋅x^ d x.(6.33)

Using Eq. (6.32) in Eq. (6.33), we obtain the result

V e m f m=w l ω B 0 sin α=A ω B 0 sin α,(6.34)

where A=w l is the surface area of the loop. The angle α is related to ω by

α=ω t+C 0,(6.35)

where C 0 is a constant determined by initial conditions. For example, if α=0 at t=0, then C 0=0. In general,

V e m f m=A ω B 0 sin(ω t+C 0) (V).(6.36)

This same result can also be obtained by applying the general form of Faraday’s law given by Eq. (6.6). The flux linking the surface of the loop is

Φ=∫S B⋅d s=∫S z ˆ B 0⋅n ˆ d s=B 0 A cos α=B 0 A cos(ω t+C 0),(6.37)

and

V e m f=−d Φ d t=−d d t[B 0 A cos(ω t+C 0)]=A ω B 0 sin(ω t+C 0),(6.38)

which is identical with the result given by Eq. (6.36).

Concept Question 6-6: Contrast the operation of an ac motor with that of an ac generator.

Concept Question 6-7: The rotating loop of Fig. 6-12 had a single turn. What would be the emf generated by a loop with 10 turns?

Concept Question 6-8: The magnetic flux linking the loop shown in Fig. 6-12 is maximum when α=0 (loop in x−y plane), yet according to Eq. (6.34), the induced emf is zero when α=0. Conversely, when α=90∘, the flux linking the loop is zero, but V e m f m is at a maximum. Is this consistent with your expectations? Why?