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 68

Chapter 28, Problem 68

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.

Verified step by step guidance

Step 1: Understand the problem. The capacitor discharges through a resistor, and the current decreases over time. This is a classic RC (resistor-capacitor) circuit problem. The current follows an exponential decay described by the equation:

I = I

₀⁢e-t/τ, where

τ = R C

is the time constant.

Step 2: Plot the data. Use the given current values and corresponding time values to create a graph. Plot the natural logarithm of the current (

\ln (I)

) versus time (

t

). This will linearize the exponential decay equation, as

\ln (I) = \ln (I

₀)-t/τ.

Step 3: Determine the slope of the linear graph. The slope of the graph (

- 1 / τ

) is related to the time constant

τ

. Use the slope to calculate the resistance

R

, knowing the capacitance

C = 20 μ F

. Rearrange the formula

τ = R C

to solve for

R

R = τ / C

Step 4: Determine the initial capacitor voltage. The initial current

I

₀ can be found from the y-intercept of the graph (

\ln (I

₀)). Use Ohm's law

V

₀=I₀⁢R to calculate the initial voltage across the capacitor.

Step 5: Verify your results. Ensure the calculated resistance and initial voltage are consistent with the data and the physical behavior of the RC circuit. Double-check the graph and calculations for accuracy.

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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 electrical charge, measured in farads (F). A 20 μF capacitor can store a charge proportional to the voltage across it, following the relationship Q = C * V, where Q is charge, C is capacitance, and V is voltage. Understanding capacitance is crucial for analyzing how the capacitor discharges through the resistor over time.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor, expressed as V = I * R. This fundamental principle is essential for determining the resistance in the circuit by analyzing the current and voltage data collected during the capacitor's discharge.

Exponential Decay in RC Circuits

In an RC (resistor-capacitor) circuit, the voltage across the capacitor and the current through the resistor decrease exponentially over time after the switch is closed. The voltage can be described by the equation V(t) = V0 * e^(-t/RC), where V0 is the initial voltage, R is resistance, and C is capacitance. This behavior is key to graphing the data and extracting both the resistance and the initial voltage of the capacitor.

Recommended video:

Guided course

04:24

Amplitude Decay in an LRC Circuit

Related Practice

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

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?

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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

The capacitor in FIGURE P28.73 begins to charge after the switch closes at t = 0 s. Find an expression for the current I at time t. Graph I from t = 0 to t = 5τ.

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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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