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where the summations are taken along any continuously directed path from p1 to p2 in the network If v1 2 comes out positive, this means that p1 is at a positive potential relative to p2
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EXAMPLE 111 Referring to Fig 111, nd I1, I2, and I3 Directions shown for the currents are arbitrary
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At junction a, I1 I2 I3 0 (1)
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Another junction equation can be written at d, but it is the negative of Eq (1) Loop Equations Traversing the loop adcba of the planar network, we obtain
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FIGURE 111 (From Wells and Slusher,2 p 237)
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100 and traversing afeda, we obtain 60
10I1
12I2
4( I3)
(It is assumed that the batteries have no internal resistance) Note that the loop afedcba would not give an independent equation, but rather the sum of Eq (2) and Eq (3) The independent equations (1), (2), and (3) can now be solved simultaneously (applying usual methods) for I1, I2, and I3, yielding I 6053 A I2 5263 A I3 07895 A
Note that each value of I is positive; thus the arbitrarily chosen directions indicated in Fig 111 happen to be correct
EXAMPLE 112
Figure 112a and b shows two possibilities, the so-called series and parallel combinations of resistors Using also Ohm s law, it can be shown that resistances in series add directly and resistances in parallel add as their reciprocals
Figure 113a and b shows two possibilities, the capacitors connected in parallel (a) and the capacitors connected in series (b) Where capacitors C1, C2, C3, are added in parallel (Fig 113a), the total capacitance C increases, that is,
EXAMPLE 113
Where capacitors C1, C2, C3 are added in series (Fig 113b), the total capacitance C decreases, that is, 1 C 1 C1 1 C2 1 C3 (2)
The capacitance C of a capacitor is the ratio of quantity of electricity q stored in it and voltage v across it, that is,
FIGURE 112
TOPICS IN APPLIED PHYSICS
FIGURE 113
Where a capacitor requires a charge of 1 C to be charged to a voltage of 1 V, its capacitance is 1 F (farad), that is, 1F 1 C V 1 A s V
9 Direct and Alternating Currents Direct current (dc) ows in one direction only through a circuit The associated direct voltages, in contrast to alternating voltages, are of unchanging polarity Direct current corresponds to a drift or displacement of electric charge in one unvarying direction around the closed loop or loops of an electric circuit Direct currents and voltages may be of constant magnitude or may vary with time Alternating current (ac) reverses direction periodically, usually many times per second One complete period, with current ow rst in one direction and then in the other, is called a cycle, and 60 cycles per second (60 hertz, or Hz) is the customary frequency of alternation in the United States and 50 Hz in Europe Alternating current is shown diagrammatically in Fig 114 Time is measured horizontally (beginning at any arbitrary moment), and the current at each instant is
FIGURE 114 Diagram of sinusoidal alternating current
CHAPTER ELEVEN
measured vertically In this diagram it is assumed that the current is alternating sinusoidally; that is, the current i is described by the following relation: i Im sin 2 t (1126)
where Im is the maximum instantaneous current, is frequency per second (Hz), and t is time in seconds Electric energy is ordinarily generated as alternating current
MAGNETIC FIELDS
1 Magnetism Magnetism comprises those physical phenomena involving mag-
netic elds and their effects on materials Magnetic elds may be set up on a macroscopic scale by electric currents or by magnets On an atomic scale, individual atoms cause magnetic elds when their electrons have a net magnetic moment as a result of their angular momentum A magnetic moment arises whenever a charged particle has an angular momentum It is the cooperative effect of the atomic magnetic moments that causes the macroscopic magnetic eld of a permanent magnet
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