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Ch 27: Current and Resistance
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 27: Current and ResistanceProblem 75
Chapter 27, Problem 75

A 300 μF capacitor is charged to 9.0 V, then connected in parallel with a 5000 Ω resistor. The capacitor will discharge because the resistor provides a conducting pathway between the capacitor plates, but much more slowly than if the plates were connected by a wire. Let t=0 s be the instant the fully charged capacitor is first connected to the resistor. At what time has the capacitor voltage decreased by half, to 4.5 V?

Verified step by step guidance
1
Identify the key components of the problem: The capacitor has a capacitance \( C = 300 \mu F = 300 \times 10^{-6} \; F \), the resistor has a resistance \( R = 5000 \; \Omega \), and the initial voltage across the capacitor is \( V_0 = 9.0 \; V \). The goal is to find the time \( t \) when the voltage across the capacitor decreases to \( V = 4.5 \; V \).
Recall the formula for the voltage across a discharging capacitor in an RC circuit: \( V(t) = V_0 e^{-t / \tau} \), where \( \tau = RC \) is the time constant of the circuit. Here, \( \tau \) represents the time it takes for the voltage to decrease to approximately 36.8% of its initial value.
Calculate the time constant \( \tau \): Substitute \( R = 5000 \; \Omega \) and \( C = 300 \times 10^{-6} \; F \) into the formula \( \tau = RC \). This gives \( \tau = 5000 \times 300 \times 10^{-6} \; s \).
Set up the equation for the voltage at time \( t \): Substitute \( V(t) = 4.5 \; V \), \( V_0 = 9.0 \; V \), and the expression for \( \tau \) into the formula \( V(t) = V_0 e^{-t / \tau} \). This gives \( 4.5 = 9.0 e^{-t / \tau} \).
Solve for \( t \): Divide both sides of the equation by \( 9.0 \), resulting in \( 0.5 = e^{-t / \tau} \). Take the natural logarithm of both sides to isolate \( t \): \( \ln(0.5) = -t / \tau \). Rearrange to find \( t \): \( t = -\tau \ln(0.5) \). Substitute the value of \( \tau \) calculated earlier to determine \( t \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage, measured in farads (F). In this scenario, the 300 μF capacitor can store a significant amount of charge at 9.0 V. The capacitance determines how much energy can be stored and how quickly it can be released when connected to a circuit.
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Capacitors & Capacitance (Intro)

RC Time Constant

The RC time constant, denoted as τ (tau), is a measure of the time it takes for the voltage across a capacitor to either charge or discharge to approximately 63.2% of its maximum value. It is calculated as τ = R × C, where R is the resistance in ohms and C is the capacitance in farads. This concept is crucial for understanding how quickly the capacitor voltage decreases over time.
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Exponential Decay

Exponential decay describes the process by which the voltage across a discharging capacitor decreases over time, following the equation V(t) = V0 * e^(-t/τ). Here, V0 is the initial voltage, e is the base of the natural logarithm, and t is time. This concept helps in determining the specific time at which the voltage drops to half its initial value, which is essential for solving the given problem.
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Related Practice
Textbook Question

You've decided to protect your house by placing a 5.0-m-tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you've learned that lightning bolts can carry up to 50 kA of current and last up to 50 μs. You don't want the potential difference between the top and bottom of the lightning rod to exceed 100 V. What minimum diameter must the rod have?

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Textbook Question

A 2.0-mm-diameter wire formed from a composite material has a resistivity that decreases with distance along the wire as ρ=ρ₀e−αx, where ρ₀=4.0×10−5 Ω m, x (in m) is measured from one end of the wire, and the constant α=4.0 m−1. What is the resistance of a 50-cm-long length of this wire?

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Textbook Question

Two 10-cm-diameter metal plates 1.0 cm apart are charged to ±12.5 nC. They are suddenly connected together by a 0.224-mm-diameter copper wire stretched taut from the center of one plate to the center of the other. Does the current increase with time, decrease with time, or remain steady? Explain.

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