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27.15: RC Circuits: Charging A Capacitor

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Physics

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RC Circuits: Charging A Capacitor
 
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27.15: RC Circuits: Charging A Capacitor

A circuit containing resistance and capacitance is called an RC circuit. A capacitor is an electrical component that stores electric charge by storing energy in an electric field. Consider a simple RC circuit having a DC (direct current) voltage source ε, a resistor R, a capacitor C, and a two-way position switch. In the circuit, the capacitor can be charged or discharged depending on the position of the switch.

When the switch is moved to connect the battery, the circuit reduces to a simple series connection of the voltage source, the resistor, the capacitor, and the switch. In this circuit, if we apply Kirchhoff's law, we obtain a differential equation that is further integrated to obtain an expression for the charge on the charging capacitor as a function of time.

Equation1

The charge on the capacitor and the current through the resistor has an inverse relationship: as the charge increases, the current decreases. The current through the resistor can be obtained by taking the time derivative of the charge.

Equation2

At time "t " equal to RC, the current in the RC circuit decreases to 0.368 of its initial value. At the same time, the capacitor charge has reached 0.632 of its maximum value. Therefore, the product RC is a measure of how quickly the capacitor charges. This term RC is called the circuit's time constant or the relaxation time.

Equation3

When the time constant is small, the capacitor charges quickly; when the time constant  is larger, the charging takes more time. If the resistance is small, it's easier for the current to flow, and the capacitor charges more quickly.


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RC Circuits Capacitor Charging Time Constant Kirchhoff's Law Current Charge Voltage Source Resistor Series Connection Differential Equation

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