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 73d

Chapter 28, Problem 73d

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.

Verified step by step guidance

Step 1: Recall the relationship for the charging current in an RC circuit. The current I(t) as a function of time t is given by the equation:

I(t) = \(\frac{\varepsilon}{R}\) e^{-t/\(\tau\)}

, where

\(\varepsilon\)

is the emf of the battery,

R

is the resistance, and

\(\tau\) = RC

is the time constant of the circuit.

Step 2: Substitute the time constant

\(\tau\) = RC

into the equation. The expression becomes:

I(t) = \(\frac{\varepsilon}{R}\) e^{-t/(RC)}

. This is the general expression for the current in the circuit as a function of time.

Step 3: To graph the current from

t = 0

t = 5\(\tau\)

, note that the current starts at its maximum value

I(0) = \(\frac{\varepsilon}{R}\)

when

t = 0

, and it decreases exponentially toward zero as time increases.

Step 4: Plot the graph of

I(t)

versus time. The x-axis represents time

t

, and the y-axis represents the current

I(t)

. The curve will start at

\(\frac{\varepsilon}{R}\)

t = 0

and approach zero asymptotically as

t

increases. Mark key points such as

t = \(\tau\)

t = 2\(\tau\)

, and so on, where the current decreases significantly.

Step 5: Interpret the graph. The exponential decay of the current reflects the charging process of the capacitor. At

t = \(\tau\)

, the current has decreased to approximately 37% of its initial value, and by

t = 5\(\tau\)

, the current is nearly zero, indicating the capacitor is almost fully charged.

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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 system to store electric charge per unit voltage. It is measured in farads (F) and is defined by the formula C = Q/V, where Q is the charge stored and V is the voltage across the capacitor. Understanding capacitance is crucial for analyzing how capacitors behave in circuits, especially during charging and discharging phases.

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 charge to approximately 63.2% of its maximum value when connected to a voltage source. It is calculated as τ = R × C, where R is the resistance in the circuit and C is the capacitance. This concept is essential for determining the charging and discharging rates of capacitors in circuits.

Current in a Charging Capacitor

The current I in a charging capacitor can be described by the equation I(t) = (V/R) * e^(-t/τ), where V is the voltage of the power source, R is the resistance, and τ is the time constant. This equation shows that the current decreases exponentially over time as the capacitor charges. Understanding this relationship is key to graphing the current over time and analyzing the behavior of the circuit.

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Textbook Question

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