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Correct answer: B. R · C
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 $R \cdot C$.
allery