An L-R-C series circuit has L = 0.450 H, C = 2.50 * 10^-5 F, and resistance R. (a) What is the angular frequency of the circuit when R = 0?

Verified step by step guidance

1. Identify the given values: Inductance (L) = 0.450 H, Capacitance (C) = 2.50 \times 10^{-5} F, and Resistance (R) = 0 \Omega.
2. Recall the formula for the angular frequency (${\omega }_0$) of an L-R-C series circuit when the resistance is zero (R = 0), which is the natural angular frequency of the circuit. The formula is ${\omega }_0=\frac{1}{\sqrt{LC}}$.
3. Substitute the values of L and C into the formula: ${\omega }_0=\frac{1}{\sqrt{0.450\times 2.50\times {10}^{-5}}}$.
4. Calculate the value inside the square root and then take the reciprocal of the square root to find ${\omega }_0$.
5. The result from the calculation will give you the angular frequency of the circuit in radians per second.

Key Concepts

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an oscillating system, such as an L-R-C circuit, oscillates in radians per second. It is related to the frequency of oscillation and is calculated using the formula ω = 2πf, where f is the frequency. In an L-R-C circuit, the angular frequency is influenced by the inductance (L) and capacitance (C) of the circuit.

Resonance in L-R-C Circuits

Resonance occurs in an L-R-C circuit when the inductive reactance equals the capacitive reactance, leading to maximum current flow. At resonance, the circuit oscillates at its natural frequency, which is determined by the values of L and C. When resistance (R) is zero, the circuit can achieve this condition more easily, allowing for a clearer calculation of angular frequency.

Inductive and Capacitive Reactance

Inductive reactance (XL) and capacitive reactance (XC) are the opposition to current flow in an inductor and capacitor, respectively. They are frequency-dependent, with XL = ωL and XC = 1/(ωC). In a series L-R-C circuit, the total reactance determines the circuit's behavior, especially at different frequencies, and is crucial for calculating the angular frequency when resistance is absent.