Understanding the Time Constant in RC Circuits

The time constant (denoted by \(τ\)) quantifies how quickly a capacitor charges or discharges in an RC (Resistor-Capacitor) circuit. It is defined as the time it takes for the voltage across the capacitor to reach approximately 63.2% of its final value during charging or to decrease to 36.8% of its initial value during discharging.

Formula for Time Constant:

\[ τ = R \times C \]

Where:

• \(τ\): Time constant (seconds)
• \(R\): Resistance (\(Ω\))
• \(C\): Capacitance (\(F\))

How It Works:

• During charging, the capacitor voltage follows an exponential curve, reaching 63.2% of its final value in one time constant (\(τ\)).
• During discharging, the capacitor voltage decreases to 36.8% of its initial value after one time constant (\(τ\)).

Voltage Across Capacitor:

For charging and discharging, the voltage is given by:

• Charging: \[ V(t) = V_{\text{final}} \left( 1 - e^{-t / τ} \right) \]
• Discharging: \[ V(t) = V_{\text{initial}} e^{-t / τ} \]

Example Calculations:

Example 1:

Given \(R = 10kΩ\) and \(C = 100µF\):

\[ τ = 10,000 \times 0.0001 = 1 \, \text{second} \]

Example 2:

Given \(R = 1MΩ\) and \(C = 10nF\):

\[ τ = 1,000,000 \times 0.00000001 = 0.01 \, \text{seconds (10ms)} \]