The equivalent capacitance of two capacitors connceted in series and in parallel are `C_s " and "C_p` respectively. Determine the capacitance of each capacitor?

Explanation

Let `C_1 " and "C_2` be the values of the two capacitances. According to the question,
`C_1+C_2= C_p" ""…………"(1)`
`(C_1C_2)/(C_1+C_2)=C_s" "………….."(2)`
From equations (1) and (2),
`(C_1C_2)/(C_p) =C_s`
`:. C_2 =(C_pC_s)/(C_1)`
Using this value in equation (1),
`C_1+(C_pC_s)/(C_1)= C_p" or, " C_1^2-C_pC_1+C_pC_s=0`
`:. C_1 = (C_p pm sqrt(C_P^2-4C_pC_s))/(2)`
Using the positive sign(+) for `C_1`, we have,
So, the two capacitances are,
`(1)/(2)[C_p+ sqrt(C_p^2-4C_pC_s)]" and "(1)/(2)[C_p - sqrt(C_p^2-4C_pC_s)]`.