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at t 0. Find the voltage across and current through the capacitor for t > 0. The time constant of the circuit is RC 0:02 s. Following the analysis of Example 7.2, we get v t 12 Ae 50t From the initial conditions, v 0 v 0 12 A 2 or A 10. v t 12 10e 50t V i t 12 v =5000 2 10 3 e 50t A 2e 50t mA The current may also be computed from i C dv=dt . And so the voltage increases exponentially from an initial value of 2 V to a nal value of 12 V, with a time constant of 20 ms, as shown in Fig. 7-4(a), while the current decreases from 2 mA to zero as shown in Fig. 7-4(b). Thus, for t > 0,

Observation cannot serve as proof. Eyesight, as in the case of a color-blind person, may be defective. Appearances may be misleading. Thus, in each part of Fig. 2-2, AB does not seem to equal CD although it actually does.

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Fig. 7-8 The dc load line, with the same intercepts as in Example 4.7, is superimposed on the characteristics of Fig. 7-8; however, because the plate characteristics are di erent from those of Example 4.7, the quiescent values are now IPQ 11:3 mA and VPQ 186 V. Then a time axis on which to plot vG 4 2 sin !t V is constructed perpendicular to the dc load line at the Q point. Time axes for iP and vP are also constructed as shown, and values of iP and vP corresponding to particular values of vG t are found by projecting through the dc load line, for one cycle of vG . The result, in Fig. 7-8, shows that vP varies from 152 to 218 V and iP ranges from 8.1 to 14.7 mA.

Fig. 7-4

Fig. 2-2

In the RL circuit of Fig. 7-5, assume that at t 0 the current is I0 . For t > 0, i should satisfy Ri L di=dt 0, the solution of which is i Aest . By substitution we nd A and s: A R Ls est 0; The initial condition i 0 A I0 . Then i t I0 e Rt=L The time constant of the circuit is L=R.

The following treatment echoes that of Section 6.2 For the usual case of negligible grid current, (4.7) degenerates to iG 0 and the grid acts as an open circuit. For small excursions (ac signals) about the Q point, iP ip and an application of the chain rule to (4.8) leads to ip iP % diP where we have de ned 1 v gm v g rp p 7:8

Measurement cannot serve as proof. Measurement applies only to the limited number of cases involved. The conclusion it provides is not exact but approximate, depending on the precision of the measuring instrument and the care of the observer. In measurement, allowance should be made for possible error equal to half the smallest unit of measurement used. Thus if an angle is measured to the nearest degree, an allowance of half a degree of error should be made. Experiment cannot serve as proof. Its conclusions are only probable ones. The degree of probability depends on the particular situations or instances examined in the process of experimentation. Thus, it is probable that a pair of dice are loaded if ten successive 7s are rolled with the pair, and the probability is much greater if twenty successive 7s are rolled; however, neither probability is a certainty.

EXAMPLE 7.4 The 12-V battery in Fig. 7-6(a) is disconnected at t 0. for all times. Find the inductor current and voltage v

2.1 Using circles to determine group relationships In (a) to (e) each letter, such as A, B, and R, represents a set or group. Complete each statement. Show how circles may be used to represent the sets or groups. (a) If A is B and B is C, then _____ . (b) If A is B and B is E and E is R, then _____ .

CHAP. 7]

R Ls 0;

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