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Difficulty: EasyAlternating Current (AC) Circuits

An RLC series circuit operating at resonance contains an inductor of inductance 0.1 H0.1\text{ H} and a capacitor of capacitance 10 μF10\ \mu\text{F}. What is the resonant angular frequency of the circuit in radians per second (rad/s\text{rad/s})?

Answer: 1000 rad/s

Answer

The resonant angular frequency of the circuit is 1000 rad/s1000\text{ rad/s}.
The resonant angular frequency ω0\omega_0 of an AC circuit is determined by the formula ω0=1LC\omega_0 = \frac{1}{\sqrt{LC}}. Substituting the given values L=0.1 HL = 0.1\text{ H} and C=10−5 FC = 10^{-5}\text{ F} gives ω0=110−6=1000 rad/s\omega_0 = \frac{1}{\sqrt{10^{-6}}} = 1000\text{ rad/s}.

Step-by-Step Solution

1
Convert given parameters to standard SI units
L=0.1 HL = 0.1\text{ H} and C=10×10−6 F=10−5 FC = 10 \times 10^{-6}\text{ F} = 10^{-5}\text{ F}.
Calculations must be performed in base SI units for dimensional consistency.
2
Calculate the resonant angular frequency using ω0=1LC\omega_0 = \frac{1}{\sqrt{LC}}
ω0=10.1×10−5=110−6=1000 rad/s\omega_0 = \frac{1}{\sqrt{0.1 \times 10^{-5}}} = \frac{1}{\sqrt{10^{-6}}} = 1000\text{ rad/s}.
At resonance, inductive reactance equals capacitive reactance (XL=XCX_L = X_C), which yields ω0=1LC\omega_0 = \frac{1}{\sqrt{LC}}.

Key Concept

Resonant Angular Frequency
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Topics

Alternating Current (AC) Circuits

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