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Chapter 26: Problem 13

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

Expert verified
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: 蟿 = R 脳 C.

Step by step solution

01

Defining the time constant of an RC circuit

The time constant, denoted as 蟿 (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.
02

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.
03

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.
04

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.
05

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.

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Most popular questions from this chapter

You want to make an ohmmeter to measure the resistance of unknown resistors. You have a battery with voltage \(\mathrm{V}_{\mathrm{emf}}=9.00 \mathrm{~V}\), a variable resistor, \(R,\) and an ammeter that measures current on a linear scale from 0 to \(10.0 \mathrm{~mA}\) a) What resistance should the variable resistor have so that the ammeter gives its full-scale (maximum) reading when the ohmmeter is shorted? b) Using the resistance from part (a), what is the unknown resistance if the ammeter reads \(\frac{1}{4}\) of its full scale?

The dead battery of your car provides a potential difference of \(9.950 \mathrm{~V}\) and has an internal resistance of \(1.100 \Omega\). You charge it by connecting it with jumper cables to the live battery of another car. The live battery provides a potential difference of \(12.00 \mathrm{~V}\) and has an internal resistance of \(0.0100 \Omega\), and the starter resistance is \(0.0700 \Omega\). a) Draw the circuit diagram for the connected batteries. b) Determine the current in the live battery, in the dead battery, and in the starter immediately after you closed the circuit.

An RC circuit has a time constant of 3.1 s. At \(t=0\), the process of charging the capacitor begins. At what time will the energy stored in the capacitor reach half of its maximum value?

Voltmeters are always connected in parallel with a circuit component, and ammeters are always connected in series. Explain why.

A battery, a resistor, and a capacitor are connected in series in an RC circuit. What happens to the current through a resistor after a long time? Explain using Kirchhoff's rules.
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