Solving Capacitor Charge Q(t) Differential Equation With Initial Conditions

In electrical engineering, understanding the behavior of circuits containing capacitors is crucial. Capacitors, components that store electrical energy in an electric field, play a vital role in various applications, from filtering signals to energy storage systems. The charge q(t) on a capacitor, which varies with time, is governed by the interplay of circuit elements such as resistors, inductors, and voltage sources. This article delves into solving a second-order linear non-homogeneous differential equation that models the charge on a capacitor in a specific circuit. Our main objective is to find the function q(t) that accurately describes the charge on the capacitor at any given time t. The differential equation provided encapsulates the dynamic behavior of the circuit, taking into account the effects of inductance, resistance, and capacitance, as well as the influence of an external sinusoidal voltage source. By solving this equation, we can predict how the capacitor will charge and discharge over time, which is essential for designing and analyzing electrical circuits and systems. This process involves several key steps, including finding the homogeneous solution, determining the particular solution, and applying initial conditions to obtain the unique solution that matches the specific circuit conditions. The solution q(t) provides valuable insights into the circuit's performance, such as the maximum charge stored, the rate of charging and discharging, and the overall stability of the system. Understanding these dynamics is crucial for optimizing circuit design and ensuring reliable operation in various electronic applications. The complexity of the differential equation reflects the intricate interactions within the circuit, highlighting the importance of a systematic approach to solving it. This article will guide you through each step, providing clear explanations and calculations to ensure a comprehensive understanding of the capacitor charge dynamics. By mastering this process, you will gain a solid foundation for tackling more complex circuit analysis problems and designing efficient and reliable electronic systems. The ability to analyze and solve such equations is a cornerstone of electrical engineering, enabling the development of innovative technologies and the optimization of existing systems. Understanding the fundamental principles behind capacitor charge dynamics is therefore essential for any aspiring electrical engineer or technician. This knowledge allows for the precise control and utilization of capacitors in a wide range of applications, from simple electronic filters to complex energy storage solutions. The sinusoidal forcing function in the differential equation adds another layer of complexity, as it introduces oscillations into the system. The interaction between the circuit's natural response and the forcing function's frequency determines the overall behavior of the capacitor charge, which can exhibit resonance phenomena under certain conditions.

We are presented with the following second-order linear non-homogeneous differential equation that describes the charge q(t) on a capacitor as a function of time t:

10 \frac{d^2q}{dt2} + 120 \frac{dq}{dt} + 1000q = 17 \sin(2t)

This equation represents a series RLC circuit driven by a sinusoidal voltage source. The coefficients in the equation correspond to the circuit's parameters: the inductance (L = 10 Henries), the resistance (R = 120 Ohms), and the capacitance (C), which is implicitly determined by the coefficient of the q term (1000). The term on the right-hand side, 17sin(2t), represents the external sinusoidal voltage source with an amplitude of 17 Volts and a frequency of 2 rad/s. Understanding the significance of each term in the equation is crucial for solving it and interpreting the results. The second derivative term, \frac{d^2q}{dt2}, represents the acceleration of the charge, which is related to the inductance in the circuit. The first derivative term, \frac{dq}{dt}, represents the current flowing through the circuit, which is related to the resistance. The term q itself represents the charge stored in the capacitor, which is related to the capacitance. The interplay between these terms determines the dynamic behavior of the circuit, including how quickly the capacitor charges and discharges, and how the circuit responds to changes in the input voltage. The sinusoidal forcing function introduces oscillations into the system, which can lead to interesting phenomena such as resonance. The amplitude and frequency of the forcing function play a crucial role in determining the circuit's response. The initial conditions provided give us additional information about the state of the capacitor at time t = 0. Specifically, we are told that the current is zero and the charge on the capacitor is 0.0005 Coulombs. These initial conditions are essential for determining the unique solution to the differential equation, as there are infinitely many solutions without them. The initial conditions essentially