22.3

Magnetic Flux

| Motional Emf and Magnetic Flux

Motional emf, as well as any other type of induced emf, can be described in terms of a concept called magnetic flux. Magnetic flux is analogous to electric flux, which deals with the electric field and the surface through which it passes (see Section 18.9 and Figure 18-32). Magnetic flux is defined in a similar way by bringing together the magnetic field and the surface through which it passes.
Now let us see how the motional emf given in Equation 22.1 can be written in terms of the magnetic flux. Figure 22-8a shows the rod used to derive Equation 22.1; it is moving through a magnetic field between a time t = 0 s and a later time t0. During this interval, it has moved a distance x0 to the right. At a later time t, the rod has moved an even greater distance x, as part b of the drawing indicates. The speed v of the rod is the distance traveled divided by the elapsed time: . Substituting this expression for v into gives

As the drawing indicates, the term x0L is the area A0 swept out by the rod in moving a distance x0, while xL is the area A swept out in moving a distance x. In terms of these areas, the emf is

Notice how the product BA of the magnetic field strength and the area appears in the numerator of this expression. The quantity BA is given the name magnetic flux and is represented by the symbol (Greek capital letter phi); thus . The magnitude of the induced emf is the change in flux divided by the time interval during which the change occurs:

In other words, the induced emf equals the time rate of change of the magnetic flux.

Figure 22-8 (a) In a time t0, the moving rod sweeps out an area A0 = x0L. (b) The area swept out in a time t is A = xL. In both parts of the figure the areas are shaded in color.

You will almost always see the previous equation written with a minus sign—namely, . The minus sign is introduced for the following reason: The direction of the current induced in the circuit is such that the magnetic force acts on the rod to oppose its motion, thereby tending to slow down the rod (see Figure 22-5). The minus sign ensures that the polarity of the induced emf sends the induced current in the proper direction so as to give rise to this opposing magnetic force.* This issue of the polarity of the induced emf will be discussed further in Section 22.5.
The advantage of writing the induced emf as is that this relation is far more general than our present discussion suggests. In Section 22.4 we will see that can be applied to all possible ways of generating induced emfs.

| A General Expression for Magnetic Flux

In Figure 22-8 the direction of the magnetic field is perpendicular to the surface swept out by the moving rod. In general, however, may not be perpendicular to the surface. For instance, in Figure 22-9 the direction perpendicular to the surface is indicated by the normal to the surface, but the magnetic field is inclined at an angle with respect to this direction. In such a case the flux is computed using only the component of the field that is perpendicular to the surface, . The general expression for magnetic flux is

| (22.2)

If either the magnitude B of the magnetic field or the angle is not constant over the surface, an average value of the product must be used to compute the flux. Equation 22.2 shows that the unit of magnetic flux is the tesla · meter2 (T · m2). This unit is called a weber (Wb), after the German physicist Wilhelm Weber (1804–1891): 1 Wb = 1 T · m2. Example 4 illustrates how to determine the magnetic flux for three different orientations of the surface of a coil relative to the magnetic field.

Figure 22-9 When computing the magnetic flux, the component of the magnetic field that is perpendicular to the surface must be used; this component is .

Problem solving insight The magnetic flux is determined by more than just the magnitude B of the magnetic field and the area A. It also depends on the angle (see Figure 22-9 and Equation 22.2).

| Example 4 | | | Magnetic Flux

A rectangular coil of wire is situated in a constant magnetic field whose magnitude is 0.50 T. The coil has an area of 2.0 m2. Determine the magnetic flux for the three orientations, , 60.0°, and 90.0°, shown in Figure 22-10.

Figure 22-10 Three orientations of a rectangular coil (drawn as an edge view) relative to the magnetic field lines. The magnetic field lines that pass through the coil are those in the regions shaded in blue.

Reasoning The magnetic flux is defined as , where B is the magnitude of the magnetic field, A is the area of the surface through which the magnetic field passes, and is the angle between the magnetic field and the normal to the surface.
Solution The magnetic flux for the three cases is:

| Graphical Interpretation of Magnetic Flux

It is possible to interpret the magnetic flux graphically because the magnitude of the magnetic field is proportional to the number of field lines per unit area that pass through a surface perpendicular to the lines (see Section 21.1). For instance, the magnitude of in Figure 22-11a is three times larger than it is in part b of the drawing, since the number of field lines drawn through the identical surfaces is in the ratio of 3:1. Because is directly proportional to B for a given area, the flux in part a is also three times larger than that in part b. Therefore, the magnetic flux is proportional to the number of field lines that pass through a surface.

Figure 22-11 The magnitude of the magnetic field in (a) is three times as great as that in (b), because the number of magnetic field lines crossing the surfaces is in the ratio of 3:1.

The graphical interpretation of flux also applies when the surface is oriented at an angle with respect to . For example, as the coil in Figure 22-10 is rotated from to 60° to 90°, the number of magnetic field lines passing through the surface (see the field lines in the regions shaded in blue) changes in the ratio of 8:4:0 or 2:1:0. The results of Example 4 show that the flux in the three orientations changes by the same ratio. Because the magnetic flux is proportional to the number of field lines passing through a surface, we often use phrases such as “the flux that passes through a surface bounded by a loop of wire.”

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.