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will use the boldface symbols B and H to denote the vector forms of B and H ; the standard typeface will represent the scalar ux density or eld intensity in a given direction
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f = qu B
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where the symbol denotes the (vector) cross product If the charge is moving at a velocity u in a direction that makes an angle with the magnetic eld, then the magnitude of the force is given by f = quB sin (162)
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(161)
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and entering at a magnetic south pole The relative strengths of the magnetic elds generated by two magnets could be depicted as shown in Figure 161 Magnetic elds are generated by electric charge in motion, and their effect is measured by the force they exert on a moving charge As you may recall from previous physics courses, the vector force f exerted on a charge of q moving at velocity u in the presence of a magnetic eld with ux density B is given by the equation
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Weaker magnetic field
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and the direction of this force is at right angles with the plane formed by the vectors B and u This relationship is depicted in Figure 162 The magnetic ux, , is then de ned as the integral of the ux density over some surface area For the simpli ed (but often useful) case of magnetic ux lines perpendicular to a cross-sectional area A, we can see that the ux is given by the following integral: =
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Stronger magnetic field
Figure 161 Lines of force in a magnetic eld
B dA
(163)
f u B
in webers (Wb), where the subscript A indicates that the integral is evaluated over the surface A Furthermore, if the ux were to be uniform over the crosssectional area A (a simpli cation that will be useful), the preceding integral could be approximated by the following expression: =B A (164)
Figure 162 Charge moving in a constant magnetic eld
Figure 163 illustrates this idea, by showing hypothetical magnetic ux lines traversing a surface, delimited in the gure by a thin conducting wire
Figure 163 Magnetic ux lines crossing a surface
Faraday s law states that if the imaginary surface A were bounded by a conductor for example, the thin wire of Figure 163 then a changing magnetic eld would induce a voltage, and therefore a current, in the conductor More
B a b (a)
16
Principles of Electromechanics
a R b Current generating a magnetic flux opposing the increase in flux due to B (b) e + i
Figure 164 Flux direction
i Right-hand rule
precisely, Faraday s law states that a time-varying ux causes an induced electromotive force, or emf, e, as follows: d (165) e= dt A little discussion is necessary at this point to explain the meaning of the minus sign in equation 165 Consider the one-turn coil of Figure 164, which forms a circular cross-sectional area, in the presence of a magnetic eld with ux density B oriented in a direction perpendicular to the plane of the coil If the magnetic eld, and therefore the ux within the coil, is constant, no voltage will exist across terminals a and b; if, however, the ux were increasing and terminals a and b were connected for example, by means of a resistor, as indicated in Figure 164(b) current would ow in the coil in such a way that the magnetic ux generated by the current would oppose the increasing ux Thus, the ux induced by such a current would be in the direction opposite to that of the original ux density vector, B This principle is known as Lenz s law The reaction ux would then point downward in Figure 164(a), or into the page in Figure 164(b) Now, by virtue of the right-hand rule, this reaction ux would induce a current owing clockwise in Figure 164(b), that is, a current that ows out of terminal b and into terminal a The resulting voltage across the hypothetical resistor R would then be negative If, on the other hand, the original ux were decreasing, current would be induced in the coil so as to reestablish the initial ux; but this would mean that the current would have to generate a ux in the upward direction in Figure 164(a) (or out of the page in Figure 164(b)) Thus, the resulting voltage would change sign The polarity of the induced voltage can usually be determined from physical considerations; therefore the minus sign in equation 165 is usually left out We will use this convention throughout the chapter In practical applications, the size of the voltages induced by the changing magnetic eld can be signi cantly increased if the conducting wire is coiled many times around, so as to multiply the area crossed by the magnetic ux lines many times over For an N-turn coil with cross-sectional area A, for example, we have the emf d (166) e=N dt Figure 165 shows an N -turn coil linking a certain amount of magnetic ux; you can see that if N is very large and the coil is tightly wound (as is usually the case in the construction of practical devices), it is not unreasonable to presume that each turn of the coil links the same ux It is convenient, in practice, to de ne the ux linkage, , as = N so that d dt (167)
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