Page 27 - Schaum's Outline of Theory and Problems of Electric Circuits

CIRCUIT CONCEPTS
16

Fig. 2-12 [CHAP. 2

dv 6 d 3
i ¼ C ¼ 60 10 ð25:0 10 tÞ¼ 1:5A
dt dt
3
p ¼ vi ¼ 37:5 10 t ðWÞ
ð t
4 2
w C ¼ pdt ¼ 1:875 10 t ðmJÞ
0
See Fig. 2-13.
4 3 2
W max ¼ð1:875 10 Þð2 10 Þ ¼ 75:0mJ
1 2 1 6 2
or W max ¼ CV max ¼ ð60:0 10 Þð50:0Þ ¼ 75:0mJ
2 2

Fig. 2-13

2.6 A 20.0-mF capacitance is linearly charged from 0 to 400 mC in 5.0 ms. Find the voltage function
and W max .
!
400 10 6 2
q ¼ t ¼ 8:0 10 t ðCÞ
5:0 10 3
3
v ¼ q=C ¼ 4:0 10 t ðVÞ
1 2
3
3
V max ¼ð4:0 10 Þð5:0 10 Þ¼ 20:0V W max ¼ CV max ¼ 4:0mJ
2
2.7 A series circuit with R ¼ 2
, L ¼ 2 mH, and C ¼ 500 mF has a current which increases linearly
from zero to 10 A in the interval 0 t 1 ms, remains at 10 A for 1 ms t 2 ms, and decreases
linearly from 10 A at t ¼ 2 ms to zero at t ¼ 3 ms. Sketch v R , v L , and v C .
v R must be a time function identical to i, with V max ¼ 2ð10Þ¼ 20 V.
For 0 < t < 1 ms,
di 3 di
¼ 10 10 A=s and v L ¼ L ¼ 20 V
dt dt
When di=dt ¼ 0, for 1 ms < t < 2 ms, v L ¼ 0.