Time constant of an RC circuit is:

In an RC (Resistor-Capacitor) circuit, the time constant characterizes the rate at which a capacitor charges or discharges through a resistor. 1. **The Differential Equation**: For a charging RC circuit, the voltage across the capacitor $V_c(t)$ is governed by the equation: $$V(t) = V_0(1 - e^{-t/RC})$$ 2. **Defining the Constant**: The term in the exponent must be dimensionless. Since $t$ has units of time, the product $RC$ must also have units of time. This product is defined as the time constant $\tau$. $$\tau = R \cdot C$$ 3. **Physical Significance**: After one time constant ($t = \tau$), the capacitor has charged to approximately $63.2\%$ of its maximum value. 4. **Unit Analysis**: - Resistance $R = \frac{V}{I} = \frac{\text{Volts}}{\text{Coulombs/Second}}$ - Capacitance $C = \frac{Q}{V} = \frac{\text{Coulombs}}{\text{Volts}}$ - Multiplying them: $\frac{\text{Volts} \cdot \text{Seconds}}{\text{Coulombs}} \cdot \frac{\text{Coulombs}}{\text{Volts}} = \text{Seconds}$. Therefore, the time constant is $R \cdot C$.