Question

Explain why the time constant for an \(\mathrm{RC}\) circuit increases with \(R\) and with \(C\). (The answer "That's what the formula says" is not sufficient.)

Short answer

Answer: The time constant for an RC circuit (τ) increases with resistance (R) and capacitance (C) due to the combined effects of R limiting the flow of current and C defining the amount of charge that can be stored. A larger resistance slows down the circuit response by limiting the current flow, while a larger capacitance takes longer to charge or discharge due to its increased ability to store charge. As a result, the time constant, τ = RC, will increase with both R and C.

Step-by-step solution

Definition of an RC circuit

An RC circuit consists of a resistor (R) and a capacitor (C) connected in series. The circuit charges or discharges through the resistor. The behavior of the circuit depends on the resistance (R) and capacitance (C).

Define the time constant

The time constant (τ) is a parameter that indicates the time it takes for the voltage across the capacitor to charge up to approximately 63.2% (1 - 1/e) of its maximum value or to discharge to approximately 36.8% of its initial value. This time constant characterizes the speed at which the capacitor charges or discharges and is influenced by both the resistor and the capacitor.

Derive the formula for the time constant

By applying Kirchhoff's Voltage Law (KVL) around the RC circuit, we get: \[V_R + V_C = V_{source}\] Since the voltage across the resistor (VR) is given by Ohm's law: \(V_R = I \cdot R\), and the voltage across the capacitor (VC) is given by \(V_C = Q/C\), where Q is the charge on the capacitor, we can write: \[IR + \frac{Q}{C} = V_{source}\] Now, the current (I) is related to the rate of change of charge (Q) by: \[I = \frac{dQ}{dt}\] Replacing I in the previous equation, we get: \[R \cdot \frac{dQ}{dt} + \frac{Q}{C} = V_{source}\] Now, if we consider the charging process, we can solve this differential equation for Q(t): \[Q(t) = C \cdot V_{source}(1 - e^{-\frac{t}{RC}})\] As we can observe, the time constant (τ) for an RC circuit is given by the product of the resistance (R) and the capacitance (C): \[\tau = RC\]

Explain the dependence on R and C

From the formula for the time constant (τ = RC), we can see that the time constant increases with both R and C. This can be explained as follows: 1. When the resistance (R) increases, it limits the flow of current through the circuit, taking more time for the capacitor to either charge or discharge. The larger the resistance, the slower the circuit response. 2. When the capacitance (C) increases, it means the capacitor can store more charge, and it takes a longer time to charge or discharge. The larger the capacitance, the more charge it can store, and thus, the slower the circuit response. In conclusion, the time constant for an RC circuit increases with the resistance (R) because it limits the flow of current through the circuit, resulting in a slower charging or discharging process. Moreover, it increases with the capacitance (C) because a larger capacitor can store more charge and thus takes a longer time to charge or discharge.