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प्रश्न
In a series LCR circuit with an AC source, R = 300 Ω, C = 20 μF, L = 1.0 henry, εrms = 50 V and ν = 50/π Hz. Find (a) the rms current in the circuit and (b) the rms potential difference across the capacitor, the resistor and the inductor. Note that the sum of the rms potential differences across the three elements is greater than the rms voltage of the source.
उत्तर
Given:
Resistance in series LCR circuit, R = 300 Ω
Capacitance in series LCR circuit, C = 20 μF= 20 × 10−6 F
Inductance in series LCR circuit, L = 1 Henry
RMS value of voltage, εrms = 50 V
Frequency of source, f = 50/ ` pi`Hz
Reactance of the inductor (XL) is given by,
`X_C = omegaL = 2pifL`
`rArr X_C = 500 Ω`
(a) Impedance of an LCR circuit (Z) is given by,
`Z = sqrt(R^2 + (X_C - X_L )^2 `
`rArr Z = sqrt((300)^2 + (500 - 100)^2`
`rArr Z = sqrt((300)^2 + (400)^2`
⇒ Z = 500
RMS value of current Irms is given
`I_{rms} = epsilon_{rms}/Z`
`rArr I_{rms} = 50/500`
`rArr I_{rms} = 0.1A`
(b) Potential across the capacitor `(V_C)` is given by,
`V_C = I_{rms} xx X_C`
`rArr V_C = 0.1xx500 = 50V`
Potential difference across the resistor `(V_R)` is given by
`V_R =I_{rms}xxR`
`V_R = 0.1xx300xx = 30V`
Potential difference across the inductor `(V_L)` is given by,
`V_L = I_{rms]xxXL`
` rArr V_L =0.1xx100 = 10V `
R.M.s potential = 50V
Net sum of all the potential drops = 50 V + 30 V + 10 V = 90 V
Sum of the potential drops > RMS potential applied
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