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) = `\V_{rms}` = 120 V


To Find:

Resonant Frequency = f₀ = ?

Root Mean Square current at f₀  = `\I_{rms}` = ?


Solution:

The formula for resonant frequency is given by:

f₀ = `\frac {1}{2ㄫ sqrt {LC}}`

by putting the corresponding values

f₀ = `\frac {1}{2 × 3.1416 sqrt {3.00 × 10⁻³ H × 5.00 ｘ10⁻⁶ F}}`

f₀ = `\frac {1}{6.283 sqrt {1.5 × 10⁻⁸ }}` Hz

f₀ = `\frac {1}{6.283 × 1.225 × 10⁻⁴}` Hz

f₀ = `\frac {1}{7.695 × 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:

`\I_{rms}` = `\frac {V_{rms}}{R}`

by putting values

`\I_{rms}` = `\frac {120 V}{40 Ω}`

`\I_{rms}` = 3 A ----------------Ans. (2)


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