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Table 161 Relative permeabilities for common materials Material Air Permalloy Cast steel Sheet steel Iron r 1 100,000 1,000 4,000 5,195
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permeability of free space, 0 = 4 10 7 H/m, times the relative permeability, r , which varies greatly according to the medium For example, for air and for most electrical conductors and insulators, r is equal to 1 For ferromagnetic materials, the value of r can take values in the hundreds or thousands The size of r represents a measure of the magnetic properties of the material A consequence of Amp` re s law is that, the larger the value of , the smaller the current required to e produce a large ux density in an electromagnetic structure Consequently, many electromechanical devices make use of ferromagnetic materials, called iron cores, to enhance their magnetic properties Table 161 gives approximate values of r for some common materials Conversely, the reason for introducing the magnetic eld intensity is that it is independent of the properties of the materials employed in the construction of magnetic circuits Thus, a given magnetic eld intensity, H, will give rise to different ux densities in different materials It will therefore be useful to de ne sources of magnetic energy in terms of the magnetic eld intensity, so that different magnetic structures and materials can then be evaluated or compared for a given source In analogy with electromotive force, this source will be termed magnetomotive force (mmf) As stated earlier, both the magnetic ux density and eld intensity are vector quantities; however, for ease of analysis, scalar elds will be chosen by appropriately selecting the orientation of the elds, wherever possible Amp` re s law states that the integral of the vector magnetic eld intensity, e H, around a closed path is equal to the total current linked by the closed path, i: H dl = i (1620)
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where dl is an increment in the direction of the closed path If the path is in the same direction as the direction of the magnetic eld, we can use scalar quantities to state that H dl = i (1621)
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Figure 169 illustrates the case of a wire carrying a current i, and of a circular path of radius r surrounding the wire In this simple case, you can see that the magnetic eld intensity, H, is determined by the familiar right-hand rule This rule states that if the direction of the current i points in the direction of the thumb of one s right hand, the resulting magnetic eld encircles the conductor in the direction in which the other four ngers would encircle it Thus, in the case of Figure 169, the closed-path integral becomes equal to H (2 r), since the path and the magnetic eld are in the same direction, and therefore the magnitude of the magnetic eld intensity is given by H = i 2 r (1622)
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Now, the magnetic eld intensity is unaffected by the material surrounding the conductor, but the ux density depends on the material properties, since B = H Thus, the density of ux lines around the conductor would be far greater in the presence of a magnetic material than if the conductor were surrounded by air The eld generated by a single conducting wire is not very strong; however, if we arrange the wire into a tightly wound coil with many turns, we can greatly increase
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i By the right-hand rule, the current, i, generates a magnetic field intensity, H, in the direction shown
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Conducting wire Circular path
Figure 169 Illustration of Amp` re s law e
the strength of the magnetic eld For such a coil, with N turns, one can verify visually that the lines of force associated with the magnetic eld link all of the turns of the conducting coil, so that we have effectively increased the current linked by the ux lines N-fold The product N i is a useful quantity in electromagnetic circuits, and is called the magnetomotive force,2 F (often abbreviated mmf), in analogy with the electromotive force de ned earlier: F = Ni ampere-turns (A t) (1623)
Figure 1610 illustrates the magnetic ux lines in the vicinity of a coil The magnetic eld generated by the coil can be made to generate a much greater ux density if the coil encloses a magnetic material The most common ferromagnetic materials are steel and iron; in addition to these, many alloys and oxides of iron as well as nickel and some arti cial ceramic materials called ferrites also exhibit magnetic properties Winding a coil around a ferromagnetic material accomplishes two useful tasks at once: it forces the magnetic ux to be concentrated near the coil and if the shape of the magnetic material is appropriate completely con nes the ux within the magnetic material, thus forcing the closed path for the ux lines to be almost entirely enclosed within the ferromagnetic material Typical arrangements are the iron-core inductor and the toroid of Figure 1611 The ux densities for these inductors are given by the expressions B= Ni l N i B= 2 r2 Flux density for tightly wound circular coil Flux density for toroidal coil (1624) (1625)
Intuitively, the presence of a high-permeability material near a source of magnetic ux causes the ux to preferentially concentrate in the high- material, rather than in air, much as a conducting path concentrates the current produced by an electric eld in an electric circuit In the course of this chapter, we shall
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