Question

A resistance of 100 `Omega`, inductor of self-inductance`(4/pi^2)` H and a capacitor of unknown capacitance are connected in series to an a.c. source of 200 V and 50 Hz. When the current and voltage are in phase, the capacitance and power dissipated is respectively ____________.

Options

• 2.5 x 10^-5 F, 400 W

• 3.0 x 10^-5 F, 50 W

• 2.0 x 10^-5 F, 100 W

• 1.5 x 10^-5 F, 200 W

Answer 1

A resistance of 100 `Omega`, inductor of self-inductance`(4/pi^2)` H and a capacitor of unknown capacitance are connected in series to an a.c. source of 200 V and 50 Hz. When the current and voltage are in phase, the capacitance and power dissipated is respectively 2.5 x 10^-5 F, 400 W.

Explanation:

`"R" = 100  Omega        "V" = 200  "V"`

`"L" = 4/pi^2 "H"     "f" = 50  "Hz "       "C" = ?`

When voltage and current are in phase then it is a pure resistive circuit.

`therefore "X"_"C" = "X"_"L"`

`1/(omega"C") = omega "L"`

`therefore omega^2 = 1/"LC"`

`therefore "C" = 1/("L"omega^2)`

`therefore "C" = 1/(4pi^2"f"^2"L"^2)`

`= 1/(4 xx pi xx 50 xx 50 xx 4/pi^2)`

`= 1/(16 xx 2500)`

`therefore "C" = 25 mu"F"`

`= 2.5 xx 10^-5 "F"`

`"I"_0 = "E"_0/"z"`

`= (200 sqrt2)/100`

`= 2 sqrt2`

`"E"_0 = 200sqrt2`

∴ Power dissipated  = E_rms I_rms

`= 200 xx 2 = 400  "W"`