Problem 2 - University Physics with Modern Physics

Question

A resistor and a capacitor are connected in series. If a second identical resistor is connected in series in the same circuit, the time constant for the circuit will a) decrease. b) increase. c) stay the same.

Step-by-Step Solution

Answer

Question: When a second identical resistor is connected in series to a resistor-capacitor circuit, the time constant will: a) Decrease b) Increase c) Remain the same d) Cannot be determined Answer: b) Increase

Step 1: Understand the time constant

The time constant (τ) is a measure of the time it takes for the capacitor to charge or discharge in a resistor-capacitor (RC) circuit, with τ = R * C. In an RC circuit, the time constant shows how long it takes the capacitor to charge up to around 63.2% of its final value or discharge down to about 36.8% of its initial value. The resistor(s) in the circuit determine the charging and discharging rate of the capacitor.

Step 2: Examine the initial circuit

Initially, the circuit has a resistor (R1) and a capacitor (C) connected in series. The time constant for this simple RC circuit is τ1 = R1 * C.

Step 3: Analyze the circuit with two identical resistors in series

Now, an identical resistor (R2) is connected in series to the circuit. The total resistance in the circuit becomes the sum of R1 and R2: R_total = R1 + R2. Since R1 = R2, we can rewrite this as R_total = 2 * R1. The time constant for this new circuit becomes τ2 = R_total * C = (2 * R1) * C.

Step 4: Compare the time constants of both circuits

To determine the effect of adding the second resistor, we need to compare the time constants of the two circuits: τ1 = R1 * C τ2 = (2 * R1) * C As we can see, τ2 is twice the value of τ1. Thus, the time constant for the circuit has increased.

Step 5: Choose the correct answer

Based on the comparison of the time constants, we can conclude that the correct answer is: b) Increase.

Key Concepts

Resistor-Capacitor Series Circuit

Imagine you're building a simple electrical circuit using just two components: a resistor and a capacitor. This setup is known as a resistor-capacitor series circuit. It's akin to connecting two beads on a string; electricity must flow through the resistor before it can reach and fill up the capacitor, much like water flowing through a pipe into a bucket.

The resistor slows down the flow of electricity, representing a form of resistance, and the capacitor stores electrical energy, temporarily holding onto it until it's needed. The series configuration ensures that the same amount of electric current that passes through the resistor also passes through the capacitor. As more resistors are added, like a series of gates before the bucket, it takes longer for the bucket (capacitor) to fill up. Consequently, the capacity of the system to charge and discharge the capacitor is affected.

Electrical Charge and Discharge

In a resistor-capacitor series circuit, two opposite yet complementary processes occur: charging and discharging. When a voltage is applied, the capacitor begins charging up, akin to water starting to fill a balloon. It fills up gradually due to the resistor's controlling effect on the flow rate, not all at once. Conversely, the discharging process is like letting go of the balloon's opening; the stored energy in the capacitor is released back into the circuit.

The rate at which these processes happen is typically not instantaneous and depends largely on the resistance and capacitance values in the circuit. A higher resistance means a slower charge and discharge process, whereas a higher capacitance requires more time to fill up or empty out completely. Understanding this dynamic helps explain how electrical energy is managed in transient situations, such as powering on or off devices.

Time Constant in RC Circuits

The heart of understanding how quickly a capacitor charges or discharges lies in the concept of the time constant, denoted as \( \tau \). This is more than just a value; it's a significant indicator of how the resistor-capacitor circuit behaves over time. The time constant \( \tau \) can be thought of as the circuit's 'rhythm,' with a simple formula \( \tau = R \times C \) where \( R \) is the resistance in ohms and \( C \) is the capacitance in farads.

Why is the time constant important? Because it tells us how long it takes for the capacitor to reach about 63.2% of the full voltage when charging, or to fall to 36.8% when discharging. In our original exercise, by doubling the resistance in the series circuit, we essentially doubled the time constant. This means it will now take longer for the capacitor to charge up to its maximum voltage and similarly longer to discharge, marking a significant change in how the circuit manages time and energy flows. Understanding this principle is key to designing circuits with predictable and desired behaviors.