For the same RLC series circuit having a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 µF capacitor: (a) Find the resonant frequency. (b) Calculate Iᵣₘₛ at resonance if Vᵣₘₛ is 120 V. Ans ( (a) 1.30 kHz, (b) 3.00 A)
Data Given:
Resistance of resistor = R = 40.0 Ω
Inductance of an inductor = L = 3.00 mH = 3.00 × 10⁻³ H
Capacitance of capacitor = C = 5.00 μF = 5.00 x10⁻⁶ F
Voltage (rms) = VrmsVrms = 120 V
To Find:
Resonant Frequency = f₀ = ?
Root Mean Square current at f₀ = IrmsIrms = ?
Solution:
The formula for resonant frequency is given by:
f₀ = 12ㄫ√LC
by putting the corresponding values
f₀ = 12×3.1416√3.00×10⁻³H×5.00x10⁻⁶F
f₀ = 16.283√1.5×10⁻⁸ Hz
f₀ = 16.283×1.225×10⁻⁴ Hz
f₀ = 17.695×10⁻⁴ Hz
f₀ = 0.12995 × 10⁴ Hz
or
f₀ = 1.2995 × 10³ Hz
or
f₀ = 1.2995 kHz -------------------Ans. (1)
Since at resonance, the RLC circuit behaves like a pure resistive circuit and the impedance Z = R, So, the rms current is given by:
Irms = VrmsR
by putting values
Irms = 120V40Ω
Irms = 3 A ----------------Ans. (2)
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