Calculate the resultant capacitances for each of the following combinations of capacitors .

Explanation

Capacitro 1 & 2 are connected in parallel
`C_(p)=C_(0)+C_(0)=2C_(0)`
Capacitor `C_(P)andC_(3)` are in series.
`(1)/(C_(s))=(1)/(C_(P))+(1)/(C_(3))implies(1)/(C_(s))=(1)/(2C_(0))+(1)/(C_(0))`
`=(1+2)/(2C_(0))=(3)/(2C_(0))`
`:.C_(s)=(2)/(3).C_(0)`
The resultant capacitance `=(2)/(3)C_(0)`
(b)
`C_(1)&C_(2)` are in series `(1)/(Cs_(1))=(1)/(C_(0))+(1)/(C_(0))=(2)/(C_(0))`
`Cs_(1)&Cs_(2)` are in parallel.
`:.C_(P)=Cs_(1)+Cs_(2)=(C_(0))/(2)+(C_(0))/(2)=C_(0)`
Resultant capacitance `=C_(0)`
(c).
Resultant capacitance `C_(P)=C_(0)+C_(0)=C_(0)`
`C_(P)=3C_(0)`
(d).
`C_(1)andC_(3)` are in series `(1)/(Cs_(1))=(1)/C_(1)+(1)/(C_(3))=(C_(1)+C_(3))/(C_(1)C_(3))`
`Cs_(1)=(C_(1)C_(3))/(C_(1)+C_(3))`
`C_(2)andC_(4)` are in series
`(1)/(Cs_(1))=(1)/(C_(2))+(1)/(C_(4))`
`Cs_(2)=(C_(2)+C_(4))/(C_(2)C_(4))=(C_(2)C_(4))/(C_(2)+C_(4))`
Now `Cs_(1)andCs_(2)` are parallel
`Cp=Cs_(1)+Cs_(2)`
`=(C_(1)C_(3))/(C_(1)+C_(3))+(C_(2)+C_(4))/(C_(2)+C_(4))`
`=(C_(1)C_(3)+C_(1)C_(3)C_(4)+C_(1)C_(2)C_(4)+C_(3)C_(2)C_(4))/((C_(1)+C_(3))(C_(2)+C_(4)))`
(e).
Capacitors 1 and 2 are in series
`(1)/(Cs_(1))=(1)/(C_(0))+(1)/(C_(0))=(2)/(C_(0))`
`Cs_(1)=(C_(0))/(2)`
Parallerly 4 and 5 are in series
`Cs_(2)=(C_(0))/(2)`
`Cs_(1),Cs_(2)3` are in parallel
`:.C_(P)=(C_(0))/(2)+(C_(0))/(2)+C_(0)`
`=C_(0)+C_(0)`
Resultant capacitance =2C