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A single-loop RLC circuit is a fundamental concept in electrical engineering, where the circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series. This type of circuit is known for its characteristic oscillatory behavior due to the interplay between the inductive and capacitive components. Understanding the behavior of a single-loop RLC circuit is crucial, as it helps in analyzing and predicting the circuit's response to changes in voltage and current over time. In this exercise, we specifically analyze a circuit with the following given components:

• Resistor: \( R = 7.20 \, \Omega \)
• Inductor: \( L = 12.0 \, \mathrm{H} \)
• Capacitor: \( C = 3.20 \, \mu \mathrm {F} \)

Additionally, the initial charge on the capacitor is \(Q_0 = 6.20 \, \mu \mathrm{C}\). A critical aspect to note is that the initial current \( I_0 = 0 \).

In RLC circuits, oscillations often occur due to the exchange of energy between the inductor and the capacitor. However, real-world resistors always dissipate some energy, leading to a gradual reduction in the amplitude of oscillations over time. This phenomenon is known as damped oscillations. The oscillations in the circuit are characterized by a decrease in the maximum charge on the capacitor and the maximum current in the circuit as energy is lost as heat in the resistor. The damping effect in the circuit is largely influenced by the resistor's value. In the provided exercise, damping needs to be taken into account to analyze the decrease in charge over multiple cycles.

Energy dissipation in an RLC circuit is primarily caused by the resistor. As the circuit oscillates, the resistor converts some of the electrical energy into thermal energy (heat), which is then lost from the system. This process is crucial for understanding how the energy initially stored in the capacitor gradually decreases over time, leading to reduced oscillation amplitudes. To quantify this energy loss, we can look at the damping factor \( \alpha \), which is calculated as \( \alpha = \frac{R}{2L} \). This parameter gives us an idea of how quickly energy is dissipated in the circuit. The greater the value of the resistor, the more significant the damping effect, causing quicker energy dissipation.

The angular frequency (\( \omega_0 \)) of the undamped oscillation in the circuit is calculated using the formula: \( \omega_0 = \frac{1}{\sqrt{LC}} \). This frequency represents the rate at which the charge on the capacitor oscillates if there were no resistive losses. However, in the presence of damping, we also need to consider the damped angular frequency \( \omega_1 \), which is given by \( \omega_1 = \sqrt{ \omega_0^2 - \alpha^2 } \). This adjusted frequency takes into account the impact of the resistor and gives a more accurate representation of the actual oscillation rate in the circuit. Both frequencies are essential for analyzing how quickly and smoothly the oscillations decrease over time.

The decay of charge in the capacitor over time due to the damping effect is an essential consideration in this exercise. The formula to calculate the charge remaining on the capacitor after \( N \) complete cycles is: \( Q_N = Q_0 e^{-\alpha NT} \left| \cos(\omega_1 NT) \right| \), where \( T \) is the period of one oscillation cycle. This equation highlights how the initial charge (\( Q_0 \)) decreases exponentially with each cycle due to the damping factor (\( \alpha \)). The cosine function accounts for the oscillatory nature of the charge. Using this formula, we can calculate the charge on the capacitor after 5, 10, and even 100 cycles, offering insights into the long-term behavior of the circuit.