An L-R-C series circuit has L = 0.400 H, C = 7.00 μF, and R = 320 Ω. At t = 0 the current is

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An L-R-C series circuit has L = 0.400 H, C = 7.00 μF, and...

An $L$-$R$-$C$ series circuit has $L = 0.400$ H, $C = 7.00 \, \mu \mathrm{F}$, and $R = 320 \, \Omega$. At $t = 0$ the current is zero and the initial charge on the capacitor is $2.80 \times 10^{-4}$ C. (a) What are the values of the constants A and $\phi$ in Eq. (30.28)? (b) How much time does it take for each complete current oscillation after the switch in this circuit is closed? (c) What is the charge on the capacitor after the first complete current oscillation?

Key Concepts

Initial Conditions and Phase Determination

When solving the differential equations for RLC circuits, initial conditions such as the initial charge on the capacitor and the initial current are essential. They allow for the determination of the specific constants in the general solution, specifically the amplitude and phase constants. This ensures that the mathematical solution accurately reflects the physical state of the circuit at the time the analysis begins.

Natural and Damped Frequency

The natural frequency of an LC circuit is determined by the inductance and capacitance (given by the formula ?? = 1/?(LC)). However, when resistance is included, the circuit oscillates at a lower, damped frequency, which accounts for the energy loss. The oscillation period of the circuit is directly related to this damped frequency, dictating the time interval between successive peaks in current or charge.

Transient Response

The transient response of an RLC circuit describes the behavior of the circuit immediately after a change, such as closing a switch. This response is characterized by decaying oscillations of current and voltage or charge until the system eventually reaches steady-state behavior. Understanding the transient response is crucial for predicting how the circuit will react to sudden inputs and how long it will take to complete one full oscillation cycle.

RLC Circuit Dynamics

An RLC series circuit combines a resistor (R), an inductor (L), and a capacitor (C) in one loop. The dynamics of such circuits are governed by a second?order differential equation derived from Kirchhoff’s voltage law, which describes how the energy is exchanged between the capacitor and inductor while the resistor dissipates energy. The system can exhibit oscillatory behavior where the charge and current vary sinusoidally over time, often with a decay due to the resistance.

Damped Oscillations

In circuits where resistance is present, the oscillations are not perpetual but are damped over time due to energy loss in the resistor. The solution to the differential equations includes an exponential decay factor that modulates the sinusoidal terms. This damping influences both the amplitude and the effective frequency of the oscillations, leading to a damped or shifted phase in the oscillatory response.

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