EBK PHYSICS FOR SCIENTISTS AND ENGINEER
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
9th Edition
ISBN: 8220100654428
Author: Jewett
Publisher: Cengage Learning US
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Textbook Question
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Chapter 32, Problem 32.26P

The switch in Figure P31.15 is open for t < 0 and is then thrown closed at time t = 0. Find (a) the current in the inductor and (b) the current in the switch as functions of time thereafter.

(a)

Expert Solution
Check Mark
To determine
The current in the inductor.

Answer to Problem 32.26P

The current in inductor in the terms of time is ε5R(1e2.5RLt) .

Explanation of Solution

Given info: The value of resistance R is 4.00Ω , inductance of the circuit is 1.00H and emf of the battery is 10.0V .

Explanation:

Formula to calculate current in a loop as per Kirchhoff law is,

i=0IsIIL=0I=IsIL (1)

Here,

Is is current flowing through switch s .

I is current flowing through resistance R .

Il is the current flowing through the inductance L .

Write the expression for net voltage in loop 1,

εRIsRI=0 (2)

Write the expression to calculate net voltage in loop 2,

εRIs2RILLdILdt=0 (3)

Here,

R is resistance of circuit.

L is inductance of circuit.

dILdt rate of change of current in inductance.

Substitute IsIL for I in equation (II).

εRIsR(IsIL)=0ε2RIsRIL=0Is=ε2R+IL2 (4)

Substitute ε2R+IL2 for Is in equation (3).

εR(ε2R+IL2)2RILLdILdt=0ε(ε2+RIL2)2RILLdILdt=0ε2RIL22RILLdILdt=0ε25RIL2LdILdt=0

Arrange the terms of above equation to simplify for integration.

LdILdt=ε22.5RILL2.5RdILdt=ε5RILL2.5RdILdt=(ε5R+IL)dIL(ε5R+IL)=2.5RLdt

On integrate,

dILε5R+IL=(2.5RL)dt (5)

Assume ε5R+IL=T .

Differentiate above equation,

dIL=dT

Substitute dT for dIL and T for ε5R+IL in equation (5).

dILT=(2.5RL)dtdILT=2.5RLdtlnT=2.5RLt+c

Substitute ε5R+IL for T in above equation,

ln(ε5R+IL)=2.5RLt+c (6)

Apply boundary condition,

Substitute 0 for t and 0 for IL in above equation.

ln(ε5R+0)=2.5RL×0+cc=ln(ε5R)

Substitute ln(ε5R) for c in equation (VI)

ln(ε5R+IL)=2.5RLt+ln(ε5R)ln(ε5R+IL)ln(ε5R)=2.5RLtln(ε5R+IL)(ε5R)=2.5RLtln(1ILε5R)=2.5RLt

Further solve the above expression.

1ILε5R=e2.5RLtIL=ε5R(1e2.5RLt)

Thus, the current in inductor in the terms of time is IL=ε5R(1e2.5RLt) .

Conclusion:

Therefore, the current in inductor in the terms of time is IL=ε5R(1e2.5RLt) .

(b)

Expert Solution
Check Mark
To determine
The current in the switch as a function of time.

Answer to Problem 32.26P

The current in the switch as the function of time is 3ε5Rε10Re2.5RLt .

Explanation of Solution

Formula to calculate current in inductor, from equation (VII)

IL=ε5R(1e2.5RLt)

Formula to calculate current in switch, from equation, from equation (IV)

Is=ε2R+IL2

Substitute ε5R(1e2.5RLt) for IL in above equation to calculate Is .

Is=ε2R+12(ε5R)(1e2.5RLt)=ε2R+ε10R(1e10.0t)=ε2R+ε10R+ε10Re10.0t=3ε5Rε10Re2.5RLt

Thus, current in switch is 3ε5Rε10Re2.5RLt .

Conclusion:

Therefore, the current in switch is 3ε5Rε10Re2.5RLt .

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Chapter 32 Solutions

EBK PHYSICS FOR SCIENTISTS AND ENGINEER

Ch. 32 - Prob. 32.6OQCh. 32 - Prob. 32.7OQCh. 32 - Prob. 32.1CQCh. 32 - Prob. 32.2CQCh. 32 - A switch controls the current in a circuit that...Ch. 32 - Prob. 32.4CQCh. 32 - Prob. 32.5CQCh. 32 - Prob. 32.6CQCh. 32 - The open switch in Figure CQ32.7 is thrown closed...Ch. 32 - After the switch is dosed in the LC circuit shown...Ch. 32 - Prob. 32.9CQCh. 32 - Discuss the similarities between the energy stored...Ch. 32 - Prob. 32.1PCh. 32 - Prob. 32.2PCh. 32 - Prob. 32.3PCh. 32 - Prob. 32.4PCh. 32 - An emf of 24.0 mV Ls induced in a 500-turn coil...Ch. 32 - Prob. 32.6PCh. 32 - Prob. 32.7PCh. 32 - Prob. 32.8PCh. 32 - Prob. 32.9PCh. 32 - Prob. 32.10PCh. 32 - Prob. 32.11PCh. 32 - A toroid has a major radius R and a minor radius r...Ch. 32 - Prob. 32.13PCh. 32 - Prob. 32.14PCh. 32 - Prob. 32.15PCh. 32 - Prob. 32.16PCh. 32 - Prob. 32.17PCh. 32 - Prob. 32.18PCh. 32 - Prob. 32.19PCh. 32 - When the switch in Figure P32.18 is closed, the...Ch. 32 - Prob. 32.21PCh. 32 - Show that i = Iiet/ is a solution of the...Ch. 32 - Prob. 32.23PCh. 32 - Consider the circuit in Figure P32.18, taking =...Ch. 32 - Prob. 32.25PCh. 32 - The switch in Figure P31.15 is open for t 0 and...Ch. 32 - Prob. 32.27PCh. 32 - Prob. 32.28PCh. 32 - Prob. 32.29PCh. 32 - Two ideal inductors, L1 and L2, have zero internal...Ch. 32 - Prob. 32.31PCh. 32 - Prob. 32.32PCh. 32 - Prob. 32.33PCh. 32 - Prob. 32.34PCh. 32 - Prob. 32.35PCh. 32 - Complete the calculation in Example 31.3 by...Ch. 32 - Prob. 32.37PCh. 32 - A flat coil of wire has an inductance of 40.0 mH...Ch. 32 - Prob. 32.39PCh. 32 - Prob. 32.40PCh. 32 - Prob. 32.41PCh. 32 - Prob. 32.42PCh. 32 - Prob. 32.43PCh. 32 - Prob. 32.44PCh. 32 - Prob. 32.45PCh. 32 - Prob. 32.46PCh. 32 - In the circuit of Figure P31.29, the battery emf...Ch. 32 - A 1.05-H inductor is connected in series with a...Ch. 32 - A 1.00-F capacitor is charged by a 40.0-V power...Ch. 32 - Calculate the inductance of an LC circuit that...Ch. 32 - An LC circuit consists of a 20.0-mH inductor and a...Ch. 32 - Prob. 32.52PCh. 32 - Prob. 32.53PCh. 32 - Prob. 32.54PCh. 32 - An LC circuit like the one in Figure CQ32.8...Ch. 32 - Show that Equation 32.28 in the text Ls Kirchhoffs...Ch. 32 - In Figure 31.15, let R = 7.60 , L = 2.20 mH, and C...Ch. 32 - Consider an LC circuit in which L = 500 mH and C=...Ch. 32 - Electrical oscillations are initiated in a series...Ch. 32 - Review. Consider a capacitor with vacuum between...Ch. 32 - Prob. 32.61APCh. 32 - An inductor having inductance I. and a capacitor...Ch. 32 - A capacitor in a series LC circuit has an initial...Ch. 32 - Prob. 32.64APCh. 32 - When the current in the portion of the circuit...Ch. 32 - At the moment t = 0, a 24.0-V battery is connected...Ch. 32 - Prob. 32.67APCh. 32 - Prob. 32.68APCh. 32 - Prob. 32.69APCh. 32 - At t = 0, the open switch in Figure P31.46 is...Ch. 32 - Prob. 32.71APCh. 32 - Prob. 32.72APCh. 32 - Review. A novel method of storing energy has been...Ch. 32 - Prob. 32.74APCh. 32 - Review. The use of superconductors has been...Ch. 32 - Review. A fundamental property of a type 1...Ch. 32 - Prob. 32.77APCh. 32 - In earlier times when many households received...Ch. 32 - Assume the magnitude of the magnetic field outside...Ch. 32 - Prob. 32.80CPCh. 32 - To prevent damage from arcing in an electric...Ch. 32 - One application of an RL circuit is the generation...Ch. 32 - Prob. 32.83CP
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What is Electromagnetic Induction? | Faraday's Laws and Lenz Law | iKen | iKen Edu | iKen App; Author: Iken Edu;https://www.youtube.com/watch?v=3HyORmBip-w;License: Standard YouTube License, CC-BY