Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

An RC circuit with a $1 Ω$ resistor and a $0.000001 - F$ capacitor is driven by a voltage $E (t) = \sin 100 tV$ . If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

The subsequent resistor is $E_{R} = - 10^{- 4} \cos 100 t + 10^{- 8} \sin 100 t - 10^{- 4} e^{- 10^{6} t}$ .

The subsequent capacitor voltage is $E_{C} = - 10^{- 4} \cos 100 t + \sin 100 t + 10^{- 4} e^{- 10^{6} t}$ .

The subsequent current is $I = - 10^{- 4} \cos 100 t + \sin 100 t + 10^{- 4} e^{- 10^{6} t}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Important formula.

The governing differential equation for RC circuit is $\frac{d q}{d t} + \frac{q}{C R} = \frac{E}{R}$ .

Step 2: Determine the subsequent resistor.

The governing differential equation for RC circuit is $\frac{d q}{d t} + \frac{q}{C R} = \frac{E}{R} . . . . . . (1)$ .

Here $q (0) = 0, C = 10^{- 6}, E (t) = \sin 100 t$ .

Put these values in equation (1) then

$\frac{d q}{d t} + 10^{- 6} q = \sin 100 t$

The integrating factor is $e^{10^{6} t}$ .

Now the equation is

$e^{10^{6} t} q = \int e^{10^{6} t} \sin 100 t d t e^{10^{6} t} q = - 10^{- 10} e^{10^{6} t} \cos 100 t + 10^{- 6} e^{10^{6} t} \sin 100 t + C q (t) = 10^{- 10} \cos 100 t + 10^{- 6} \sin 100 t + C e^{- 10^{6 t}}$

Now, find the value of done

$I = \int e^{10^{6} t} \sin 100 t d t = - \frac{1}{100} e^{10^{6} t} \cos 100 t + \frac{10^{6}}{100} \int e^{10^{6} t} \cos 100 t d t = - \frac{1}{100} \cos 100 t + 1000 \int e^{10^{6} t} \cos 100 t d t = - \frac{1}{100} e^{10^{6} t} \cos 100 t + 1000 (\frac{1}{100} e^{10^{6} t} \sin 100 t - \frac{10^{6}}{100} \int e^{10^{6} t} \sin 100 t d t) = - \frac{1}{100} e^{10^{6} t} \cos 100 t + 100 (\frac{1}{100} e^{10^{6} t} \sin 100 t - \frac{10^{6}}{100} - 10000 I = - \frac{1}{100} e^{10^{6} t} \cos 100 t + 100 e^{10^{6} t} \sin 100 t - 100000000 I + C I = - 10^{- 10} e^{10^{6}} \cos 100 t + 10^{- 6} \sin 100 t + C$

Apply the initial conditions then $C = - 10^{- 10}$ .

$I = - 10^{- 10} e^{10^{6}} \cos 100 t + 10^{- 6} \sin 100 t + - 10^{- 10} q (t) = 10^{- 10} \cos 100 t + 10^{- 6} \sin 100 t - 10^{- 10} e^{- 10^{6} t}$

The subsequent resistor is $E_{C} = - 10^{- 4} \cos 100 t + \sin 100 t +' 10^{- 4} e^{- 10^{6} t}$ .

Step 3: evaluate capacitor voltage.

Now, find the value of capacitor voltage.

$E_{C} = \frac{q (t)}{C} E_{C} = \frac{10^{- 10} \cos 100 t + 10^{- 6} \sin 100 t - 10^{- 10} e^{- 10^{6} t}}{10^{- 10}} E_{C} = - 10^{- 4} \cos 100 t + \sin 100 t +' 10^{- 4} e^{- 10^{6} t}$

The subsequent capacitor voltage is $E_{C} = - 10^{- 4} \cos 100 t + \sin 100 t +' 10^{- 4} e^{- 10^{6} t}$

Step 4: Determine electric resistor.

Using Kirchhoff’s voltage law to the RC circuit

$E_{R} = E_{C} - E (t) E_{R} = - 10^{- 4} \cos 100 t + 10^{- 8} \sin 100 t - 10^{- 4} e^{- 10^{6} t}$

Step 5: find the value of current.

Now, find the value of current.

$I = \frac{E_{R}}{R} I = - 10^{- 4} \cos 100 t + \sin 100 t + 10^{- 4} e^{- 10^{6} t}$

Therefore, the subsequent current is $I = - 10^{- 4} \cos 100 t + \sin 100 t + 10^{- 4} e^{- 10^{6} t}$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

An RC circuit with a $1 Ω$ resistor and a $0.000001 - F$ capacitor is driven by a voltage $E (t) = \sin 100 tV$ . If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

Step by Step Solution

Step 1: Important formula.

Step 2: Determine the subsequent resistor.

Step 3: evaluate capacitor voltage.

Step 4: Determine electric resistor.

Step 5: find the value of current.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

An RC circuit with a 1Ω resistor and a 0.000001-F capacitor is driven by a voltage E(t)=sin100tV. If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

Step by Step Solution

Step 1: Important formula.

Step 2: Determine the subsequent resistor.

Step 3: evaluate capacitor voltage.

Step 4: Determine electric resistor.

Step 5: find the value of current.

Most popular questions for Math Textbooks

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An RC circuit with a $1 Ω$ resistor and a $0.000001 - F$ capacitor is driven by a voltage $E (t) = \sin 100 tV$ . If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.