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 increases with resistance and capacitance because a larger resistance slows down the flow of charge in the circuit, while a larger capacitance requires more time to charge or discharge due to its increased charge storage capacity. This is reflected in the formula for the time constant, which is the product of resistance and capacitance: τ = R × C.

Step-by-step solution

Defining the time constant of an RC circuit

The time constant, denoted as τ (tau), of an RC circuit is a measure of the time it takes for the voltage across the capacitor to charge (or discharge) to approximately 63.2% of its final value when a step voltage is applied across the circuit. The time constant is crucial in determining the speed at which an RC circuit responds to changes in voltage.

Formula for time constant

The time constant for an RC circuit is given by the product of the resistance (R) and the capacitance (C) of the circuit: \(\tau = R \times C\). This formula shows that the time constant is directly proportional to both the resistance and the capacitance.

Understanding the role of resistance in an RC circuit

The resistance in an RC circuit determines how easily charge flows through the circuit. A larger resistance will impede the flow of charge more than a smaller resistance. This means that, for a given capacitance, it takes more time for the capacitor to charge (or discharge) when the resistance is high.

Understanding the role of capacitance in an RC circuit

The capacitance of an RC circuit determines how much charge can be stored on the capacitor for a given voltage. A larger capacitance means that more charge can be stored, and vice versa. Hence, for a given resistance in the circuit, it will take more time for a capacitor with a larger capacitance to charge (or discharge) because it has to store more charge.

Combining the effects of resistance and capacitance on the time constant

When resistance and capacitance both increase, their combined effect on the time constant is also increased. Since the time constant is the product of resistance and capacitance, an increase in either of these values will result in an increase in the time constant. This is consistent with our understanding that a larger resistance slows down the flow of charge in the circuit, while a larger capacitance requires more time to charge or discharge due to its increased charge storage capacity.