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Ch 28: Fundamentals of Circuits
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 28: Fundamentals of CircuitsProblema 72
Capitolo 28, Problema 72

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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Step 1: Start by recalling the formula for the energy stored in a capacitor: \( E = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage across the capacitor. This will help us understand how the energy changes over time.
Step 2: Use the fact that the energy decreases by 50% in 0.25 seconds. The energy at any time \( t \) during discharge is given by \( E(t) = E_0 e^{-\frac{t}{RC}} \), where \( E_0 \) is the initial energy, \( R \) is the resistance, and \( C \) is the capacitance. Substitute \( E(t) = 0.5 E_0 \) and \( t = 0.25 \) s into this equation.
Step 3: Simplify the equation \( 0.5 E_0 = E_0 e^{-\frac{0.25}{RC}} \). Cancel \( E_0 \) on both sides to get \( 0.5 = e^{-\frac{0.25}{RC}} \). Take the natural logarithm of both sides to isolate \( \frac{0.25}{RC} \): \( \ln(0.5) = -\frac{0.25}{RC} \).
Step 4: Rearrange the equation \( \ln(0.5) = -\frac{0.25}{RC} \) to solve for \( R \): \( R = -\frac{0.25}{C \ln(0.5)} \). Substitute \( C = 15 \mu F = 15 \times 10^{-6} \) F into the equation.
Step 5: Perform the substitution and simplify the expression for \( R \). Ensure the units are consistent throughout the calculation. The resistance \( R \) can now be calculated using the given values.

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Concetti chiave

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Capacitance

Capacitance is the ability of a capacitor to store electrical energy in an electric field, measured in farads (F). It is defined as the charge stored per unit voltage across the capacitor. In this case, the 15 μF capacitor can store energy based on its capacitance and the voltage applied, which is crucial for understanding how much energy is initially stored.
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Percorso guidato
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Capacitors & Capacitance (Intro)

Energy Stored in a Capacitor

The energy (E) stored in a capacitor can be calculated using the formula E = 1/2 CV², where C is the capacitance and V is the voltage. This relationship is essential for determining how much energy is present in the capacitor before and after discharge. In this scenario, knowing that the energy decreases by 50% allows us to calculate the initial and final energy values.
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Percorso guidato
09:26
Energy Stored by Capacitor

RC Time Constant

The RC time constant (τ) is a measure of the time it takes for the voltage across a capacitor to charge or discharge through a resistor. It is calculated as τ = R × C, where R is the resistance and C is the capacitance. This concept is vital for understanding the rate of energy loss in the circuit, particularly since the problem states that the energy decreases by 50% in a specific time frame.
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Percorso guidato
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Phase Constant of a Wave Function
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