What is the equivalent capacitance of the capacitor network shown in Fig. 24-7? (All the...
What is the equivalent capacitance of the capacitor network shown in Fig. 24-7? (All the capacitors have a capacitance C = 5 μF. The voltage of the battery isV = 6V.)Answer
| 3 μF | ||
| 5 μF | ||
| 10 μF | ||
| 20 μF | ||
| 1 μF |
Homework Answers
Qtotal= Q1+Q4=Q5+Q2
Q1+Q3=Q2
Q2+Q3=Q1
Q1-Q2=Q2-Q1
Q1=Q2
?Q3=0
?V3=0
V2+V3=V5 ----> V3=0
?V2=V5
V3+V4=V1 ----> V3=0
?V4=V1
V2+V1=Vtotal
V4+V5=Vtotal
Combining Equations:
Q1+Q3=Q2
Q2+Q3=Q1
?Q1=Q2
Qtotal=Q1+Q4=Q5+Q2
?Q4=Q5
?C4V4=C5V5
C1V1=C2V2
?C1/C2=V2/V1
C4/C5=V5/V4
combining equations:
V2=V5,V4=V1
?V2/V1=V5/V4
?C1/C2=C4/C5
beginning with:
CtotalVtotal=Qtotal
CtotalVtotal=Q1+Q4
Ctotal=(C1V1+C4V4)/(V1+V2)
Because V4=V1 and V2=V1C1/C2
Ctotal=((C1+C4)V1)/(V1+(V1C1/C2))
Ctotal=(C1+C4)/(1+(C1/C2))
Q.E.D.
THIS IS ONE OF 4 EQUIVALENT EXPRESSIONS FOR CTOTAL ALL OF WHICHWILL CONTAIN ONLY THREE CAPACITOR VALUES BECAUSE OF THE BALANCINGNATURE OF THE C3 CAPACITORIN THE NETWORK. We believe that theposted solution to P38 in the homework problem section of the siteis incorrect because it models the system as if C3 isnot there. Itwill carry no charge, but it will reduce the total capacitance ofthe network just as any series capacitor would.
Our answer to B isless by a factor of 1/1.75, which we feel is correct.
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