For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Q_o on the capacitor.

검증된 단계별 안내

Start by recalling the total energy stored in an LRC circuit, which is the sum of the energy stored in the electric field of the capacitor \( U_E \) and the energy stored in the magnetic field of the inductor \( U_B \). The total energy is given by \( U = U_E + U_B \).

The energy stored in the capacitor is \( U_E = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage across the capacitor. Since \( V = \frac{Q}{C} \), this can be rewritten as \( U_E = \frac{1}{2} \frac{Q^2}{C} \), where \( Q \) is the charge on the capacitor.

The energy stored in the inductor is \( U_B = \frac{1}{2} L I^2 \), where \( L \) is the inductance and \( I \) is the current through the inductor. Using the relationship \( I = -\frac{dQ}{dt} \), this becomes \( U_B = \frac{1}{2} L \left( \frac{dQ}{dt} \right)^2 \).

For an underdamped LRC circuit, the charge on the capacitor as a function of time is given by \( Q(t) = Q_0 e^{-\gamma t} \cos(\omega_d t) \), where \( Q_0 \) is the initial charge, \( \gamma = \frac{R}{2L} \) is the damping coefficient, and \( \omega_d = \sqrt{\frac{1}{LC} - \gamma^2} \) is the damped angular frequency. Substitute this expression for \( Q(t) \) into the formulas for \( U_E \) and \( U_B \).

Combine \( U_E \) and \( U_B \) to express the total energy \( U(t) \) as a function of time. Simplify the expression to show how the energy decays over time due to the damping factor \( e^{-2\gamma t} \). The final formula will involve \( Q_0 \), \( C \), \( L \), \( \gamma \), and \( \omega_d \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Underdamped LRC Circuit

An underdamped LRC circuit is a type of electrical circuit that consists of an inductor (L), a resistor (R), and a capacitor (C) where the resistance is low enough that the system oscillates rather than quickly settling to equilibrium. In this scenario, the circuit exhibits oscillatory behavior with a gradually decreasing amplitude, allowing for the analysis of energy transfer between the electric and magnetic fields over time.

Energy in Electric and Magnetic Fields

In an LRC circuit, energy is stored in two forms: electric energy in the capacitor (UE) and magnetic energy in the inductor (UB). The electric energy is given by UE = 1/2 * C * V^2, where V is the voltage across the capacitor, while the magnetic energy is given by UB = 1/2 * L * I^2, where I is the current through the inductor. The total energy U = UE + UB varies over time as energy oscillates between these two forms.

Initial Charge (Qo) on the Capacitor

The initial charge Qo on the capacitor is a critical parameter that influences the behavior of the LRC circuit. It determines the initial voltage across the capacitor and, consequently, the initial energy stored in the electric field. As the circuit oscillates, this initial charge plays a significant role in calculating the time-dependent energy stored in both the electric and magnetic fields, allowing for the derivation of a formula for U as a function of time.