There are two circuits shown in figure . Find the ratio of power factor of circuit -I to that with power factor of circuit -II .

Explanation

Impedance of this circuit can be written as follows :
`Z= sqrt(R^(2)+(X_(L)-X_(C))^(2))=sqrt(R^(2)+(3R-R)^(2))`
`implies Z=Rsqrt(5)`
Hence power factor for the circuit -I can be written as follows :
`cos delta = (R )/(Z) = ( R)/(Rsqrt(5))`
`implies cos delta= (1)/(sqrt(5))`
Circuit -II
Imepdance of this circuit can be written as follows :
`Z= sqrt(R^(2)+(X_(L))^(2))=sqrt(R^(2)+(3R)^(2))`
`implies Z =Rsqrt(10)`
Hence power factor of the circuit -II can be written as follows :
`cos delta= (R)/(Z )= (R)/(Rsqrt(10))`
`implies cos delta=(1)/(sqrt(10))`
Hence ratio of power factor can be written as follows :
`(cosdelta)/(cosdelta)= ((1)/(sqrt(5)))/((1)/(sqrt(10)))`
`(cosdelta)/(cosdelta)=(sqrt(2))/(1)`