Problem 31 - Advanced Engineering Mathematics

Question

A 100-volt electromotive force is applied to an \(R C\)-series circuit in which the resistance is 200 ohms and the capacitance is \(10^{-4}\) farad. Find the charge \(q(t)\) on the capacitor if \(q(0)=0\). Find the current \(i(t)\).

Step-by-Step Solution

Answer

The charge is \( q(t) = -\frac{1}{100}e^{-50t} + \frac{1}{100} \) and the current is \( i(t) = \frac{1}{2}e^{-50t} \).

Step 1: Set up the differential equation

The RC circuit is modeled by the differential equation \( R \frac{dq}{dt} + \frac{1}{C}q = E(t) \), where \( R = 200 \) ohms, \( C = 10^{-4} \) farad, and \( E(t) = 100 \) volts. Substituting these values gives:\[ 200 \frac{dq}{dt} + 10^4 q = 100 \]

Step 2: Simplify the differential equation

Divide through by 200 to simplify the equation:\[ \frac{dq}{dt} + 50q = \frac{1}{2} \]

Step 3: Solve the homogeneous equation

The homogeneous equation is \( \frac{dq}{dt} + 50q = 0 \). The solution is:\[ q_h(t) = Ce^{-50t}, \] where \( C \) is a constant.

Step 4: Solve the non-homogeneous equation

Use an integrating factor or guess a particular solution of the form \( q_p(t) = A \). Substitute into the differential equation:\[ A = \frac{1}{2} \cdot \frac{1}{50} = \frac{1}{100} \].

Step 5: General solution and apply initial condition

The general solution is:\[ q(t) = q_h(t) + q_p(t) = Ce^{-50t} + \frac{1}{100} \]. Apply the initial condition \( q(0) = 0 \) to find \( C \):\[ 0 = C \cdot 1 + \frac{1}{100} \Rightarrow C = -\frac{1}{100} \]. Therefore, the solution is\[ q(t) = -\frac{1}{100}e^{-50t} + \frac{1}{100}. \]

Step 6: Find the current \( i(t) \)

Current is the rate of change of charge over time, \( i(t) = \frac{dq}{dt} \). Differentiate the solution for \( q(t) \):\[ i(t) = \frac{d}{dt}\left(-\frac{1}{100}e^{-50t} + \frac{1}{100}\right) = 1/2 e^{-50t}. \]

Key Concepts

RC Circuit

An RC circuit, or resistor-capacitor circuit, is a simple electric circuit that incorporates a resistor and a capacitor. It is a crucial element in understanding electric circuits, especially in analyzing how voltages and currents change over time. In this context, the circuit is driven by a constant electromotive force (EMF).

RC circuits are commonly used to model and study transient behaviors in electronic systems. These circuits are governed by a first-order linear differential equation, relating the voltage across the capacitor, the resistance, and the capacitance.

Key components of an RC circuit include:

• Resistance (R): impedes the flow of electric current and is measured in ohms (\(\Omega\).
• Capacitance (C): the ability of a system to store charge per unit voltage, measured in farads (F).
• Charge (q): the amount of electric charge stored on the capacitor at any given time.

The behavior of these circuits over time can be profoundly understood using differential equations, making RC circuits a fundamental subject for students of electrical engineering and electronics.

Electromotive Force

Electromotive force (EMF) is a crucial concept in understanding electric circuits, such as the RC circuit. It is the potential difference that drives electric charge around a circuit, essentially acting as a "voltage source."

In an RC circuit, the EMF is what starts the process of charging the capacitor. For this problem, the EMF is specified as a constant value of 100 volts.

Some important points about EMF include:

• EMF is commonly delivered by batteries or other power sources.
• It represents the energy provided per charge that moves through the circuit.
• In the context of the problem, it's denoted by \(E(t)\), where \(t\) would represent time.

Understanding how EMF interacts with the components of the circuit, especially when there are resistors and capacitors, helps predict how the capacitor charges over time. It also aids in exploring the time-dependent characteristics of the electrical flow, captured by differential equations derived for the circuit.

Initial Conditions

Initial conditions are used in solving differential equations to determine specific solutions from general ones. They provide the necessary information to resolve unknown constants that appear during the equation-solving process.

In the examined RC circuit problem, the initial condition states that the charge on the capacitor at time \( t = 0 \) is zero, denoted as \( q(0) = 0 \). This condition allows us to find the particular constant associated with the solution of the differential equation.

Significance of initial conditions includes:

• Ensures uniqueness of the solution for differential equations.
• Guides how the system transitions from its initial state to subsequent states.
• Aligns mathematical models with physical systems; in this case, indicating that the capacitor starts uncharged.

Knowing the initial conditions helps confirm the practical behavior of the circuit's transient response and correctly predict the time evolution of both the charge \( q(t) \) and the current \( i(t) \). Initial conditions are essential when modeling real-life electronic systems.