For a resistance R and capacitance C in series the impedence is twice that of a parallel combinations of the same elements. The frequency of the applied emf shall be

Explanation

`Z_(s)=sqrt(R^(2)+X_(c)^(2))=[R^(2)+((1)/(omegaC))^(2)]^(1//2)`
In case of parallel combination`I_(R)=(V//R)sinomegatandI_(c)=(V//X_(c))sin(omegat+(pi)/(2))`
or`,I=I_(R)+I_(C)=i_(0)sin(omegat+phi)`
With `l_(o) cosphi = V//R and I_(0) sin phi= V//X_(c)`
So,`I_(0)=[((V)/(R))^(2)]^(1//2)=(V)/(Z_(p))`i.e.,`(1)/(Z_(p))=[(1)/(R^(2))+((1)/(X_(c)))^(2)]^(1//2)`,i.e.`,Z_(p)=(R)/(sqrt(1+omega^(2)C^(2)R^(2)))`,
and as according to given problem,`Z_(s)=sqrt(R^(2)+X_(c)^(2))=[R^(2)+((1)/(omegaC))^(2)]^(1//2)`
In case of parallel combination`I_(R)=(V//R)sinomegatandI_(c)=(V//X_(c))sin(omegat+(pi)/(2))`
`or,I=I_(R)+I_(C)=i_(0)sin(omegat+phi)`
With `l_(o) cosphi = V//R and I_(0) sin phi= V//X_(c)`
So,`I_(0)=[((V)/(R))^(2)]^(1//2)=(V)/(Z_(p))`i.e.,`(1)/(Z_(p))=[(1)/(R^(2))+((1)/(X_(c)))^(2)]^(1//2)`,i.e.,`Z_(p)=(R)/(sqrt(1+omega^(2)C^(2)R^(2)))`
`Z_(S)=2Z_(p)I.e>,Z_(S)^(2)=4Z_(p)^(2)`
`i.e.((R^(2)omega^(2)C^(2)+1))/(omega^(2)C^(2))=4(R^(2))/(1+R^(2)omega^(2)C^(2))
`i.e.(1+R^(2)omega^(2)C^(2))^(2)=4R^(2)omega^(2)c^(2)`
`or,1+R^(2)omega^(2)C^(2)=2RomegaC`,
`(RomegaC-1)^(2)=0`
`oromega=(1)/(RC),i.e.,f=(1)/(2piRC)`