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Chapter 29, Problem 74PQ

(a)

To determine

The expression for the time constant for this circuit.

(a)

Expert Solution
Check Mark

Answer to Problem 74PQ

The expression for the time constant is (R1R2R1+R2)C_.

Explanation of Solution

Write the expression for the equivalent resistance for this circuit as.

    Req=R1R2R1+R2                                                                                                (I)

Here, Req is the equivalent resistance of circuit, R1 is the resistance of first element and R2 is the resistance of the second element...

Write the expression for the time constant for the circuit parameter.

    τ=ReqC                                                                                                      (II)

Here, τ is the time constant and C is the capacitor of the circuit.

Conclusion:

Substitute R1R2R1+R2 for Req in equation (II).

  τ=(R1R2R1+R2)C

Thus, the expression for the time constant is (R1R2R1+R2)C_.

(b)

To determine

The expression for the time when the capacitor has lost half of its charge.

(b)

Expert Solution
Check Mark

Answer to Problem 74PQ

The expression for the time when the capacitor has lost half of its charge is (R1R2R1+R2(C))ln(2)_.

Explanation of Solution

Write the expression for the charge store in capacitor as.

    Vc(t)=Q(t)C                                                                                               (III)

Here, Vc is the voltage across the capacitor, Q is the charge store in capacitor and C is the capacitance of the capacitor

Write the expression for the initial charge stored in capacitor as.

    V0=Q0C                                                                                                      (IV)

Here, Qo is the initial charge stored in the circuit.

Write the expression for voltage across capacitor as a function of time as.

    Vc(t)=V0et/τ                                                                                               (V)

Here, V0 is the initial voltage on the capacitor, Vc(t) is the voltage after time t, R is the resistance, C is capacitance and t is time.

Conclusion:

Substitute Q(t)C for Vc(t) and Q0C for V0 in equation (V).

    Q(t)C=Q0Cet/τQ(t)=Q0et/τ

Substitute Q02 for Q(t) and ReqC for τ in the above equation.

  Q02=Q0et/ReqC12=et/ReqCln(12)=tReqC

Substitute R1R2R1+R2 for Req in above equation.

  t=(R1R2R1+R2(C))ln(2)

Thus, the expression for the time when the capacitor has lost half of its charge is (R1R2R1+R2(C))ln(2)_.

(c)

To determine

The expression for the current through the capacitor at that time.

(c)

Expert Solution
Check Mark

Answer to Problem 74PQ

The expression for the current through the capacitor at that time is (R1+R2)V0(R1R2)(12)_.

Explanation of Solution

Write the expression current through the capacitor at time t as.

    I(t)=V0Reqet/τ                                                                                         (VI)

Conclusion:

Substitute (R1R2R1+R2(C))ln(2) for t, and ReqC for τ,

  I(t)=V0Reqe((R1R2R1+R2(C))ln(2))/(ReqC)=V0Reqe((R1R2R1+R2)ln(2))/(Req)

Substitute R1R2R1+R2 for Req.in above equation.

  I(t)=V0(R1R2R1+R2)e((R1R2R1+R2)ln(2))/(R1R2R1+R2)=V0(R1R2R1+R2)eln(2)=(R1+R2)V0(R1R2)(12)

Thus, the expression for the current through the capacitor at that time is (R1+R2)V0(R1R2)(12)_.

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Chapter 29 Solutions

Webassign Printed Access Card For Katz's Physics For Scientists And Engineers: Foundations And Connections, 1st Edition, Single-term

Ch. 29 - Prob. 5PQCh. 29 - Prob. 6PQCh. 29 - A real battery (modeled as an ideal emf device in...Ch. 29 - Prob. 8PQCh. 29 - Two circuits made up of identical ideal emf...Ch. 29 - Prob. 10PQCh. 29 - Prob. 11PQCh. 29 - Prob. 12PQCh. 29 - Eight real batteries, each with an emf of 5.00 V...Ch. 29 - Prob. 14PQCh. 29 - Prob. 15PQCh. 29 - Prob. 16PQCh. 29 - Prob. 17PQCh. 29 - Prob. 18PQCh. 29 - Prob. 19PQCh. 29 - An ideal emf device with emf is connected to two...Ch. 29 - Prob. 21PQCh. 29 - Prob. 22PQCh. 29 - Prob. 23PQCh. 29 - Prob. 24PQCh. 29 - Prob. 25PQCh. 29 - Prob. 26PQCh. 29 - Determine the currents through the resistors R2,...Ch. 29 - The emf devices in the circuits shown in Figure...Ch. 29 - Prob. 29PQCh. 29 - Prob. 30PQCh. 29 - Prob. 31PQCh. 29 - Prob. 32PQCh. 29 - Prob. 33PQCh. 29 - Prob. 34PQCh. 29 - A Figure P29.35 shows a combination of six...Ch. 29 - A Each resistor shown in Figure P29.36 has...Ch. 29 - Each resistor shown in Figure P29.36 has a...Ch. 29 - Prob. 38PQCh. 29 - Prob. 39PQCh. 29 - The emf in Figure P29.40 is 4.54 V. The...Ch. 29 - Figure P29.41 shows three resistors (R1 = 14.0 ,...Ch. 29 - Figure P29.42 shows five resistors and two...Ch. 29 - The emfs in Figure P29.43 are 1 = 6.00 V and 2 =...Ch. 29 - Prob. 44PQCh. 29 - Figure P29.45 shows five resistors connected...Ch. 29 - Figure P29.46 shows a circuit with a 12.0-V...Ch. 29 - Two ideal emf devices are connected to a set of...Ch. 29 - Two ideal emf devices are connected to a set of...Ch. 29 - Three resistors with resistances R1 = R/2 and R2 =...Ch. 29 - Prob. 51PQCh. 29 - Prob. 52PQCh. 29 - Prob. 53PQCh. 29 - Prob. 55PQCh. 29 - At time t = 0, an RC circuit consists of a 12.0-V...Ch. 29 - A 210.0- resistor and an initially uncharged...Ch. 29 - Prob. 58PQCh. 29 - A real battery with internal resistance 0.500 and...Ch. 29 - Figure P29.60 shows a simple RC circuit with a...Ch. 29 - Prob. 61PQCh. 29 - Prob. 62PQCh. 29 - Prob. 63PQCh. 29 - Ralph has three resistors, R1, R2, and R3,...Ch. 29 - Prob. 65PQCh. 29 - An ideal emf device is connected to a set of...Ch. 29 - Prob. 67PQCh. 29 - An ideal emf device (24.0 V) is connected to a set...Ch. 29 - Prob. 69PQCh. 29 - What is the equivalent resistance between points a...Ch. 29 - A capacitor with initial charge Q0 is connected...Ch. 29 - Prob. 73PQCh. 29 - Prob. 74PQCh. 29 - Prob. 75PQCh. 29 - Prob. 76PQCh. 29 - Figure P29.77 shows a circuit with two batteries...Ch. 29 - In the RC circuit shown in Figure P29.78, an ideal...Ch. 29 - Prob. 79PQCh. 29 - Calculate the equivalent resistance between points...Ch. 29 - In Figure P29.81, N real batteries, each with an...Ch. 29 - Prob. 82PQCh. 29 - Prob. 83PQCh. 29 - Prob. 84PQCh. 29 - Figure P29.84 shows a circuit that consists of two...Ch. 29 - Prob. 86PQCh. 29 - Prob. 87PQCh. 29 - Prob. 88PQCh. 29 - Prob. 89PQCh. 29 - Prob. 90PQCh. 29 - Prob. 91PQCh. 29 - Prob. 92PQCh. 29 - Prob. 93PQCh. 29 - Prob. 94PQCh. 29 - Prob. 95PQ
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