Laboratory Activity: Exploring Differential Equations in RC Circuits Using Wolfram Alpha

Objective: Understand and solve differential equations for basic RC (Resistor-Capacitor) circuits using Wolfram Alpha.

Instructions for Students:

Part 1: Introduction to RC Circuits and Wolfram Alpha

1. Understanding RC Circuits:
An RC circuit includes a resistor (R) and a capacitor (C). The key to these circuits is how the capacitor charges and discharges through the resistor.
The basic differential equation for an RC charging circuit is:
$\frac{dV}{dt} = \frac{1}{RC}(V_s - V)$
Here, $V$ is the voltage across the capacitor, $V_s$ is the source voltage, $R$ is the resistance, and $C$ is the capacitance.

2. Getting Started with Wolfram Alpha:
Visit the Wolfram Alpha website.
You’ll use this tool to input the differential equation and analyze its solution.

Part 2: Simple RC Circuit Analysis

1. First Exercise – Charging of a Capacitor:
Consider a simple RC circuit with a 12V battery (so $V_s = 12$ volts), a 2Ω resistor, and a 1F capacitor.
Write and input the differential equation for this circuit into Wolfram Alpha:
$\frac{dV}{dt} = \frac{1}{2 \times 1}(12 - V)$
Observe the solution provided by Wolfram Alpha, focusing on how the voltage across the capacitor changes over time.

2. Plotting the Solution:
Use Wolfram Alpha to graph the voltage across the capacitor over time.

3. Analysis Questions:
Estimate the time for the capacitor to reach about 63% of the source voltage.
What are the effects of changing resistance or capacitance values on the charging curve?

Part 3: Extended Application

1. Second Exercise – Discharging of a Capacitor:
Consider the same circuit, but the capacitor is initially fully charged, and the battery is disconnected, leaving only the capacitor and the resistor.
The differential equation for the discharging process is:
$\frac{dV}{dt} = -\frac{1}{RC}V$
Use Wolfram Alpha to solve this equation with the given circuit parameters and plot the voltage across the capacitor over time.

2. Comparative Analysis:
Compare the charging and discharging curves and discuss any similarities or differences.

Part 4: Report Writing

1. Writing Your Findings:
Compile your observations, analysis, and plots into a concise report.
Explain how the differential equation models the RC circuit’s behavior in both charging and discharging scenarios.

Learning Outcomes:
– Learn the role of differential equations in modeling RC circuits.
– Gain skills in using computational tools like Wolfram Alpha for solving and analyzing equations.
– Develop the ability to interpret and report on electrical circuit behavior.