A series RLC circuit with ε₀= 10.0 V consists of a 1.0 Ω resistor, a 1.0 μH inductor, and a 1.0 μF capacitor. What is V₁ at ω = ω₀ and at ω = ω₁?

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Identify the key components of the series RLC circuit: a resistor (R = 1.0 Ω), an inductor (L = 1.0 μH = 1.0 × 10⁻⁶ H), and a capacitor (C = 1.0 μF = 1.0 × 10⁻⁶ F). The source voltage is ε₀ = 10.0 V.

Determine the resonant angular frequency (ω₀) of the circuit using the formula:

ω

₀=1LC. Substitute the values of L and C to calculate ω₀.

At resonance (ω = ω₀), the inductive reactance (

X

_L=ωL) and capacitive reactance (

X

_C=1ωC) cancel each other out, leaving only the resistor. Use Ohm's law (

V

₁=IR) to find V₁, where I is the current.

For ω = ω₁ (a frequency other than resonance), calculate the total impedance (Z) of the circuit using the formula:

Z = \sqrt{R^{2} + (^{X}}

_L-X_C)2. Substitute the values of R, XL, and XC to find Z.

Once Z is determined for ω = ω₁, use Ohm's law again to find V₁:

V

₁=IR, where the current I is given by

I = ε

₀Z. Substitute the values to calculate V₁.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

RLC Circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. The behavior of the circuit is influenced by the values of these components, particularly in response to alternating current (AC) signals. The circuit can exhibit resonance, where the inductive and capacitive reactances cancel each other out, leading to maximum voltage across the components.

Resonant Frequency (ω₀)

The resonant frequency (ω₀) of an RLC circuit is the frequency at which the inductive reactance equals the capacitive reactance, resulting in maximum current flow and minimal impedance. It is calculated using the formula ω₀ = 1/√(LC), where L is the inductance and C is the capacitance. At this frequency, the circuit can store and transfer energy efficiently between the inductor and capacitor.

Voltage Across Components (V₁)

The voltage across a specific component in an RLC circuit, denoted as V₁, can vary depending on the frequency of the applied voltage source. At resonance (ω = ω₀), the voltage across the components can be significantly higher than the source voltage due to the energy oscillation between the inductor and capacitor. At other frequencies (ω = ω₁), the voltage distribution changes, influenced by the impedance of the circuit.