Question

A series RLC circuit has resistance \(R\), inductance \(L\), and capacitance \(C\). At what time does the energy in the circuit reach half of its initial value?

Short answer

Question: Determine the time at which the energy in a series RLC circuit reaches half of its initial value. Answer: To find the time at which the energy in a series RLC circuit reaches half of its initial value, follow these steps: 1. Write expressions for energy stored in the inductor (W_L) and capacitor (W_C). 2. Write expressions for current (I) and voltage across the capacitor (V_C) using Kirchoff's law and Ohm's law. 3. Solve the second-order linear differential equation for charge (Q). 4. Calculate the total energy (W) in terms of charge and current expressions. 5. Find the time by setting W(1/2) = 1/2 W(0) and solving for time. The exact solution depends on the specific values of resistance (R), inductance (L), and capacitance (C).

Step-by-step solution

Write expressions for energy stored in inductor and capacitor

Inductive energy, \(W_L\), is given by the formula: \(W_L = \frac{1}{2}LI^2\), where \(I\) is the current flowing through the inductor. Similarly, capacitive energy, \(W_C\), can be given by the formula: \(W_C = \frac{1}{2}C{V_C}^2\), where \(V_C\) is the voltage across the capacitor. The total energy in the circuit, \(W\), is the sum of the inductive and capacitive energies: \(W = W_L + W_C\).

Write expressions for current and voltage across the capacitor

Kirchoff's loop law tells us that the sum of voltage changes around a closed circuit must equal zero. In our RLC circuit, we have: \(V_R + V_L + V_C = 0\). From Ohm's law, we know the voltage across the resistor is \(V_R = IR\), and the voltage across the inductor is \(V_L = L\frac{dI}{dt}\), and from the definition of capacitance, \(V_C = \frac{Q}{C}\), where \(Q\) is the charge stored in the capacitor. So, the Kirchoff's loop law equation becomes: \(IR + L\frac{dI}{dt} + \frac{Q}{C} = 0\). Now, we know that \(I = \frac{dQ}{dt}\), so we can rewrite the above equation as: \(R\frac{dQ}{dt} + L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0\). This is a second-order linear differential equation for charge \(Q\).

Solve the differential equation for charge \(Q\)

In general, the solution for the differential equation can take the form of damped oscillations. But the precise form of the solution depends on the damping factor defined by \(\zeta = \frac{R}{2\sqrt{LC}}\). For this problem, let's assume that the solution looks like: \(Q(t) = Q_0e^{-\alpha t}\cos(\omega t + \phi)\), where \(Q_0\), \(\alpha\), \(\omega\), and \(\phi\) are constants. Substitute this expression into the differential equation and solve for the constants. This step can be quite involved and depends on the specific values of \(R\), \(L\), and \(C\).

Calculate the total energy in terms of charge

Now, we calculate \(I(t) = \frac{dQ}{dt}\) using the expression for \(Q(t)\). Substitute these expressions into \(W_L = \frac{1}{2}LI^2\) and \(W_C = \frac{1}{2}C{V_C}^2\), and compute the total energy \(W = W_L + W_C\) as a function of time.

Find the time at which energy is half the initial energy

We finally need to determine when the energy reaches half of its initial value, i.e., \(W\left(\frac{1}{2}\right) = \frac{1}{2}W(0)\). This equation involves the constants \(\alpha\), \(\omega\), and \(\phi\) found earlier, and solving for the time. The exact solution depends on the specific values of \(R\), \(L\), and \(C\). To summarize, the steps to find the time at which the energy in a series RLC circuit reaches half of its initial value are: 1. Write expressions for energy stored in inductor and capacitor 2. Write expressions for current and voltage across the capacitor using Kirchoff's law and Ohm's law 3. Solve the second-order linear differential equation for charge \(Q\) 4. Calculate the total energy in terms of charge and current expressions 5. Find the time at which energy is half the initial energy by setting \(W\left(\frac{1}{2}\right) = \frac{1}{2}W(0)\) and solving for time

Key concepts

Kirchhoff's Loop Law

In electrical circuits, Kirchhoff's loop law is a fundamental principle that helps us understand how voltages behave around closed loops. This law is based on the concept of energy conservation and states that the total voltage around any closed loop in a circuit must equal zero.
This means that the sum of all individual voltage drops or rises in a loop must cancel each other out.

• For a series RLC circuit (consisting of a resistor, an inductor, and a capacitor), the loop equation can be expressed as: \(V_R + V_L + V_C = 0\).
• Here, \(V_R = IR\) represents the voltage across the resistor, \(V_L = L\frac{dI}{dt}\) is the voltage across the inductor, and \(V_C = \frac{Q}{C}\) is the voltage across the capacitor.

The voltage terms integrate Ohm's law and definitions related to inductance and capacitance. By rearranging the loop equation, it's possible to derive significant insights into the behavior and dynamics of an RLC circuit as it pertains to energy and current flow.

Inductive and Capacitive Energy

The energy storage within an RLC circuit is composed of the energies held by the inductor and the capacitor. Each component stores energy in a unique way:

• Inductive Energy: This energy is associated with the magnetic field generated by current flowing through an inductor. It is given by the expression \(W_L = \frac{1}{2}LI^2\), where \(L\) is the inductance and \(I\) is the current.
• Capacitive Energy: In contrast, capacitive energy is related to the electric field between the plates of a capacitor. It is calculated by \(W_C = \frac{1}{2}C{V_C}^2\), where \(C\) is the capacitance and \(V_C\) is the voltage.

Together, the total energy in the RLC circuit at any moment is simply the sum of these two energies: \(W = W_L + W_C\).
Understanding how energy transfers between the inductor and the capacitor is crucial for analyzing the time dynamics of the whole system, like at the point when the energy is half of its initial value.

Second-Order Linear Differential Equation

Analyses of dynamics in RLC circuits often lead to solving a second-order linear differential equation. These equations arise in many physical systems and are characterized by involving second derivatives. In the context of RLC circuits:

• We express the governing equation as \(R\frac{dQ}{dt} + L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0\), where \(Q\) is the charge on the capacitor.
• This equation corresponds to Kirchoff's loop law and represents how the charge flows through the circuit components.
• Its solution usually involves determining parameters like the damping factor and natural frequency, which depend on \(R\), \(L\), and \(C\) values.

Mathematically, solving such an equation can suggest oscillatory solutions, indicating that the circuit can exhibit behaviors similar to mechanical oscillations, such as in a spring-mass system.

Damped Oscillations

Damped oscillations describe a dying waveform where the amplitude of oscillations decreases over time. In RLC circuits, damped oscillations arise due to the resistive losses that gradually reduce the circuit's energy.

• The damping factor \(\zeta\), defined as \( \zeta = \frac{R}{2\sqrt{LC}} \), plays a pivotal role in determining the nature of these oscillations.
• Depending on \(\zeta\), the circuit can experience different damping regimes: underdamped, critically damped, or overdamped.
• In the underdamped scenario, the circuit oscillates with decreasing amplitude, while in critically damped and overdamped cases, the system returns to equilibrium without oscillating.

The relevance of damped oscillations in this context provides insight into when the energy of an RLC circuit will have decayed to half its initial value, which can be traced through the relationship between amplitude, time, and damping factor.