The impedance of a series LCR circuit is 8 ohm when `v = 60 Hz` at resonance. At `v = 80 Hz`, the impedance is 10 ohm. Calculate the values of inductance L and capacitance C.

Explanation

At resonance, `Z = R = 8 ohm, v = 60 Hz`
As `v = (1)/(2 pi sqrt(LC))`
`L = (1)/(4 pi^(2) v^(2) C)`
Again, at `v' = 80 Hz, Z = 10 ohm`
As `Z^(2) = R^(2) + (X_(L) - X_(C )^(2))`
`X_(L) - X_(C ) = sqrt(Z^(2) - R^(2)) = sqrt(10^(2) - 8^(2))`
`oemga' L (1)/(omega' C) = 6`
`2 pi (80) L - (1)/(2 pi (80) C) = 6`
Using (i), we get
`2 pi (80) xx (1)/(4 pi^(2) v^(2) C) - (1)/(2 pi (80) C) = 6`
`(80)/(2 pi(60)^(2) C) - (1)/(160 pi C) = 6`
`(1)/(C ) [(1)/(90 pi) - (1)/(160 pi)] = 6`
`(70 pi)/(90 pi xx 160 pi) = 6 C`
`C = (70)/(90 xx 160 pi xx 6) = 0.00027 F`
From (i), `L = (1)/(4 xx pi^(2) xx (60)^(2) xx 0.00027)`
`L = 0.026 H`