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0. Math Review31m
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1. Intro to Physics Units1h 29m
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6. Intro to Forces (Dynamics)3h 22m
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21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

27. Resistors & DC Circuits

Solving Resistor Circuits

Physics

27. Resistors & DC Circuits

Solving Resistor Circuits

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6:55 minutes

Problem 73d

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.

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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Related Practice

Textbook Question

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]

(II) What is the net resistance of the circuit connected to the battery in Fig. 26–46?

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Textbook Question

(III) Determine the net resistance in Fig. 26–61 (a) between points a and c, and (b) between points a and b. Assume R' ≠ R. [Hint: Apply an emf between the two points in each case and determine currents; use symmetry at junctions.]

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Textbook Question

Consider two unequal resistors, of resistance R₁ and R₂, that are connected either in series or in parallel. Fill in the Table below assuming the electric potential on the low-voltage end of the combination is V_A volts and the potential at the high-voltage end of the combination is V_B volts. First draw diagrams.

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Textbook Question

Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery ε = 15.000 V with a 10.00-MΩ resistor in series with an unknown resistor R. As shown in Fig. 26–92, a particular voltmeter reads V₁ = 366 mV when connected across the 10.00 -MΩ resistor and this meter reads V₂ = 7.317 V when connected across R. Determine the value of R. [Hint: Define R_V as the voltmeter’s internal resistance.]

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Textbook Question

(II) How long does it take for the energy stored in a capacitor in a series RC circuit (Fig. 26–65) to reach 75% of its maximum value? Express answer in terms of the time constant $τ = R C$ .

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Textbook Question

A copper pipe has an inside diameter of 3.00 cm and an outside diameter of 5.00 cm (Fig. 25–39). What is the resistance to a dc current of a 12.0-m length of this pipe?.

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Textbook Question

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?

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Textbook Question

What is the time constant for the discharge of the capacitors in FIGURE EX28.34?

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