I’m studying for my Calculus class and need an explanation.
In this part of the project, you will be modelling the behavior of a resistor-capacitor (RC) circuit that is being charged by a DC voltage source.
In this circuit, initially, the capacitor is uncharged. When switch ‘A’ closes, the voltage source (ε) begins charging the capacitor (C) through the resistor (R). Applying Kirchhoff’s Voltage Law results in the equation
Using the definition of capacitance, , where ‘q’ is the charge on the capacitor and ‘VC’ is the voltage across the capacitor, we can solve for the voltage as . Next, using Ohm’s law, the voltage drop across the resistor is . The current through any portion of the circuit is defined as the rate of change of the charge: . Inserting these values into our equation above, we have
This differential equation can be solved to determine an expression for the charge on the capacitor as a function of time:
where is the final charge on the capacitor and is known as the time constant