Ch 28: Fundamentals of Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 28: Fundamentals of Circuits

Problem 67

Chapter 28, Problem 67

The capacitor in an RC circuit is discharged with a time constant of 10 ms. At what time after the discharge begins are (a) the charge on the capacitor reduced to half its initial value and (b) the energy stored in the capacitor reduced to half its initial value?

Verified step by step guidance

Step 1: Understand the concept of the time constant (τ) in an RC circuit. The time constant is defined as τ = RC, where R is the resistance and C is the capacitance. It represents the time it takes for the charge or current to decay to approximately 37% of its initial value.

Step 2: Use the formula for the charge on a capacitor during discharge: Q(t) = Q₀ * e^(-t/τ), where Q₀ is the initial charge, t is the time, and τ is the time constant. To find the time when the charge is reduced to half its initial value, set Q(t) = Q₀/2 and solve for t.

Step 3: Rearrange the equation Q₀/2 = Q₀ * e^(-t/τ) to isolate t. Divide both sides by Q₀, resulting in 1/2 = e^(-t/τ). Take the natural logarithm (ln) of both sides to solve for t: ln(1/2) = -t/τ.

Step 4: Solve for t using the relationship t = -τ * ln(1/2). Substitute τ = 10 ms into the equation to find the time when the charge is reduced to half its initial value. Note that ln(1/2) is equivalent to -ln(2).

Step 5: For part (b), recall that the energy stored in a capacitor is proportional to the square of the charge: U(t) = (1/2) * C * [Q(t)]². To find the time when the energy is reduced to half its initial value, set U(t) = U₀/2 and solve for t using the same exponential decay formula for Q(t). This will involve solving for t in the equation Q(t) = Q₀/√2.

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Key Concepts

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

Time Constant

The time constant (τ) in an RC circuit is a measure of the time it takes for the charge or voltage to decay to approximately 37% of its initial value. It is calculated as τ = R × C, where R is the resistance and C is the capacitance. In this case, a time constant of 10 ms indicates that after this duration, the charge on the capacitor will have decreased significantly, providing a basis for understanding the discharge process.

Exponential Decay

The discharge of a capacitor follows an exponential decay model, described by the equation Q(t) = Q0 e^(-t/τ), where Q(t) is the charge at time t, Q0 is the initial charge, and e is the base of the natural logarithm. This means that the charge decreases rapidly at first and then more slowly over time, which is crucial for determining when the charge reaches half its initial value.

Energy Stored in a Capacitor

The energy (U) stored in a capacitor is given by the formula U = 1/2 C V^2, where C is the capacitance and V is the voltage across the capacitor. As the capacitor discharges, both the charge and voltage decrease, leading to a reduction in stored energy. Understanding how energy relates to charge and voltage is essential for calculating when the energy is halved during the discharge process.

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Textbook Question

A 12 V car battery dies not so much because its voltage drops but because chemical reactions increase its internal resistance. A good battery connected with jumper cables can both start the engine and recharge the dead battery. Consider the automotive circuit of FIGURE P28.64. How much current is the dead battery alone able to drive through the starter motor?

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Textbook Question

What is the current through the 10 Ω resistor in FIGURE P28.61? Is the current from left to right or right to left?

Textbook Question

A 15 μF capacitor charged to 12 V is discharged through a resistor. The energy stored in the capacitor decreases by 50% in 0.25 s. What is the value of the resistance?

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Textbook Question

How much current flows through the bottom wire in FIGURE P28.66, and in which direction?

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Textbook Question

A 150 μF defibrillator capacitor is charged to 1500 V. When fired through a patient’s chest, it loses 95% of its charge in 40 ms. What is the resistance of the patient’s chest?

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Textbook Question

A circuit you’re using discharges a 20 μF capacitor through an unknown resistor. After charging the capacitor, you close a switch at t = 0 s and then monitor the resistor current with an ammeter. Your data are as follows: Use an appropriate graph of the data to determine (a) the resistance and (b) the initial capacitor voltage.

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