BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A series circuit consists of a resistor with R = 20 Ω,  an inductor with L= 1 H, a capacitor with C = 0.002 F, and a 12-V battery. If the initial charge and current are both 0, find the charge and current at time t.

To determine

To find: The charge of series RLC circuit at time t and the current of series RLC circuit at time t .

Explanation

Given data:

Series RLC circuit with following values.

R=20Ω , L=1H , C=0.002F , E(t)=12V , Q(0)=0 and Q(0)=I(0)=0 .

Formula used:

Write the expression for differential equation of electric circuit.

Ld2Qdt2+RdQdt+1CQ=E(t)

LQ+RQ+1CQ=E(t) (1)

Write the expression for general solution with complex roots.

Q(t)=eαt[c1cos(βt)+c2sin(βt)] (2)

Write the expression for r .

r=α+βi (3)

Write the expression for auxiliary equation.

ar2+br+c=0 (4)

Write the expression for differential equation.

ay+by+cy=0 (5)

Find the differential equation for electric circuit using equation (1).

Substitute 1 for L, 20 for R, 0.002 for C and 12 for E(t) in equation (1),

1Q+20Q+10.002Q=12

Q+20Q+500Q=12 (6)

Modify equation (5) as follows.

aQ+bQ+cQ=0 (7)

Compare equation (6) and (7).

a=1b=20c=500

Substitute 1 for a, 20 for b and 500 for c in equation (4),

r2+20r+500=0

Find the value of r .

r=20±(20)24(1)(500)2(1)=20±40020002=20±16002=20±40i2

Simplify r as follows.

r=10±20i (8)

Compare equations (3) and (8).

α=10β=20

Substitute 10 for α and 20 for β in equation (2),

Q(t)=e10t[c1cos(20t)+c2sin(20t)] (9)

Modify equation (9) for complementary equation as follows.

Qc(t)=e10t[c1cos(20t)+c2sin(20t)]

Consider the value of Qp(t) as follows.

Qp(t)=A (10)

Differentiate equation (10) with respect to t .

Qp(t)=0

Differentiate equation with respect to t .

Qp(t)=0

Modify equation (6) as follows.

Qp(t)+20Qp(t)+500Qp(t)=12

Substitute 0 for Qp(t) , 0 for Qp(t) and A for Qp(t) ,

1(0)+20(0)+500(A)=12500(A)=12A=12500A=3125

Substitute 3125 for A in equation (10),

Qp(t)=3125

Write the expression for general solution.

Q(t)=Qc(t)+Qp(t)

Substitute e10t[c1cos(20t)+c2sin(20t)] for Qc(t) and 3125 for Qp(t) ,

Q(t)=e10t[c1cos(20t)+c2sin(20t)]+3125 (11)

Substitute 0 for t ,

Q(0)=e10(0)[c1cos(20×0)+c2sin(20×0)]+3125=c1+3125

Substitute 0 for Q(0) ,

0=c1+3125c1=03125c1=3125

Write the expression for I

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 17 Solutions

Show all chapter solutions add
Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Global Defense Spending Global defense spending stood at 1.44 trillion in 2009 and is projected to grow at the ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Simplify: 112144

Elementary Technical Mathematics

In problems 23-58, perform the indicated operations and simplify. 31.

Mathematical Applications for the Management, Life, and Social Sciences

Divide the following fractions and reduce to lowest terms. 23.

Contemporary Mathematics for Business & Consumers

If f(x) = sin 2x, an upper bound for |f(n + 1)(x)| is 2 2n 2n + 1 22n

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: f(x) = 3x x3 is concave down for x 1.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th