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Chapter 26: Problem 86

An \(R\) -C circuit has a time constant \(R C\) (a) If the circuit is discharging, how long will it take for its stored energy to be reduced to 1/e of its initial value? (b) If it is charging, how long will it take for the stored energy to reach 1\(/ e\) of its maximum value?

Short Answer

Expert verified
The time required is the time constant \( RC \) in both discharging and charging scenarios.

Step by step solution

01

Understanding the Time Constant

In an R-C circuit, the time constant \( \tau \) is given by \( \tau = R \times C \), where \( R \) is resistance and \( C \) is capacitance. The time constant represents the time it takes for the charge or energy to decrease to about 37% \( (1/e) \) of its initial value or to increase to about 63% of its maximum value during charging or discharging.
02

Discharging Circuit Analysis

When the circuit is discharging, the energy stored in the circuit decreases over time. The exponential decay of the energy \( E \) in the circuit can be described by the equation: \( E(t) = E_0 \cdot e^{-t/RC} \). We want \( E(t) = \frac{1}{e}E_0 \). Solving \( \frac{E_0}{e} = E_0 \cdot e^{-t/RC} \) gives \( t = RC \).
03

Charging Circuit Analysis

When the circuit is charging, the energy in the circuit increases over time. The energy \( E \) increases towards its maximum value which is \( E_{max} \). We use the equation: \( E(t) = E_{max}(1 - e^{-t/RC}) \). We seek \( E(t) = \frac{1}{e}E_{max} \). Solving \( \frac{E_{max}}{e} = E_{max}(1 - e^{-t/RC}) \) results in the same \( t = RC \).
04

Conclusion

In both cases, discharging and charging, the circuit will take a duration equal to the time constant \( RC \) for the energy to change to \( \frac{1}{e} \) of its initial or maximum value, respectively. This is due to the properties of the exponential growth and decay functions that govern R-C circuits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Constant
The concept of a time constant is pivotal in understanding R-C circuits. In this context, the time constant, denoted as \( \tau \), is defined as \( \tau = R \times C \), where \( R \) is the resistance, and \( C \) is the capacitance of the circuit. This constant measures how quickly the circuit responds to changes in voltage.
  • During discharging, it reflects the time taken for stored energy to fall to about 37% \((1/e)\) of its initial value.
  • In charging, it indicates the time taken for the energy to reach approximately 63% of its maximum capacity.
Overall, the time constant provides a simple and effective tool to understand the timing characteristics of energy changes in an R-C circuit.
Exponential Decay
In R-C circuits, the concept of exponential decay describes how stored energy diminishes over time when discharging. This process is modeled mathematically by the function \( E(t) = E_0 \cdot e^{-t/RC} \), where \( E_0 \) is the initial energy, and \( E(t) \) is the energy at time \( t \).
  • The equation implies that as the circuit discharges, the energy decreases at an exponentially diminishing rate.
  • At \( t = RC \), the energy becomes \( E_0/e \), a tipping point where only about 37% of the starting energy remains.
This exponential behavior simplifies analyzing how quickly energy dissipates and helps predict the time needed for significant energy reduction.
Energy Storage
Energy storage in an R-C circuit is a dynamic interplay of electrical parameters like resistance and capacitance. These parameters define how efficiently a circuit can store energy. Initially, when a voltage is applied to an R-C circuit, the capacitor begins to charge.
  • As the capacitor charges, it stores energy, which can be utilized later when the circuit discharges.
  • The maximum energy storage capability is determined by the capacitor's ability to hold charge and the resistance affecting the current flow.
Understanding energy storage dynamics provides insight into how R-C circuits operate and adapt under different electrical conditions.
Charging and Discharging
Charging and discharging are two critical phases of operation in an R-C circuit. Charging involves raising the energy level within the circuit as the capacitor fills with charge. This process can be described using the function \( E(t) = E_{max}(1 - e^{-t/RC}) \), where \( E_{max} \) is the maximum energy.
  • During charging, energy approaches \( E_{max} \) but needs a time constant \( RC \) to reach \( E_{max}/e \).
  • Conversely, in discharging, the process lowers the energy as the capacitor releases stored charge, adhering to exponential decay principles.
Both charging and discharging processes are fundamental to the functioning of an R-C circuit, underscoring the importance of time constants in determining circuit behavior.

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Most popular questions from this chapter

Power Rating of a Resistor. The power rating of a resistor is the maximum power the resistor can safely dissipate without too great a rise in temperature and hence damage to the resistor. (a) If the power rating of a \(15-\mathrm{k} \Omega\) resistor is \(5.0 \mathrm{W},\) what is the maximum allowable potential difference across the terminals of the resistor? (b) \(\mathrm{A} 9.0\) -k\Omega resistor is to be connected across a \(120-\mathrm{V}\) potential difference. What power rating is required? (c) A \(100.0-\Omega\) and a 150.0 -\Omega resistor, both rated at 2.00 \(\mathrm{W}\) , are cunnected in series across a variable potential difference. What is the greatest this potential difference can be without overheating either resistor, and what is the rate of heat generated in each resistor under these conditions?

A Three-Way Light Bulb. A three-way light bulb has three brightness settings (low, medium, and high) but only two filaments. (a) A particular three-way light bulb connected across a 120 -V line can dissipate \(60 \mathrm{W}, 120 \mathrm{W},\) or 180 \(\mathrm{W}\) . Describe how the two filaments are arranged in the bulb, and calculate the resistance of each filament. (b) Suppose the filament with the higher resistance burns out. How much power will the bulb dissipate on each of the three brightness settings? What will be the brightness (low, medium, or high) on each setting? (c) Repeat part (b) for the situation in which the filament with the lower resistance burns out.

An emf source with \(\mathcal{E}=120 \mathrm{V},\) a resistor with \(R=\) \(80.0 \Omega,\) and a capacitor with \(C=4.00 \mu F\) are connected in series. As the capacitor charges, when the current in the resistor is 0.900 \(\mathrm{A}\) . what is the magnitude of the charge on each plate of the capacitor?

\(120-\mathrm{V}\) circuit that has a \(20-\mathrm{A}\) circuit breaker. You plug an electric hair dryer into the same outlet. The hair dryer has power settings of \(600 \mathrm{W}, 900 \mathrm{W}, 1200 \mathrm{W},\) and 1500 \(\mathrm{W}\) . You start with the hair dryer on the \(600-\mathrm{W}\) setting and increase the power setting until the circuit breaker trips. What power setting caused the breaker to trip?

A resistor \(R_{1}\) consumes electrical power \(P_{1}\) when connected to an emf \(\mathcal{E}\) When resistor \(R_{2}\) is connected to the same emf, it consumes electrical power \(P_{2}\) . In terms of \(P_{1}\) and \(P_{2},\) what is the total electrical power consumed when they are both connected to this emf source (a) in parallel and (b) in series?
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