Fundamentals of Electric Circuits
Fundamentals of Electric Circuits
6th Edition
ISBN: 9780078028229
Author: Charles K Alexander, Matthew Sadiku
Publisher: McGraw-Hill Education
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Chapter 1, Problem 3P

Find the charge q(t) flowing through a device if the current is:

  1. (a) i(t) = 3 A, q(0) = 1 C
  2. (b) i(t) = (2t + 5) mA, q(0) = 0
  3. (c) i(t) = 20 cos(10t + π/6) μA, q(0) = 2μC
  4. (d) i(t) = 10e‒30t sin 40t A, q (0) = 0

(a)

Expert Solution
Check Mark
To determine

Calculate the charge q(t) flowing through a device when the current is i(t)=3A, and q(0)=1C.

Answer to Problem 3P

The charge q(t) flowing through the device is (3t+1)C_.

Explanation of Solution

Given data:

The current flowing through the device [i(t)] is 3 A.

The charge at initial state [q(0)] is 1 C.

Formula used:

Write the expression for charge entering to the device as follows:

q(t)=i(t)dt+q(0) (1)

Here,

i(t) is the current through the device, and

q(0) is the charge at initial state.

Calculation:

Substitute 3 A for i(t) and 1 C for q(0) in equation (1) to obtain the charge flowing through the device as follows:

q(t)=(3A)dt+1C=3tC+1C=(3t+1)C

Conclusion:

Thus, the charge q(t) flowing through the device is (3t+1)C_.

(b)

Expert Solution
Check Mark
To determine

Calculate the charge q(t) flowing through a device when the current is i(t)=(2t+5)mA, and q(0)=0mC.

Answer to Problem 3P

The charge q(t) flowing through the device is (t2+5t)mC_.

Explanation of Solution

Given data:

The current flowing through the device [i(t)] is (2t+5)mA.

The charge at initial state [q(0)] is 0mC.

Calculation:

Substitute (2t+5)mA for i(t) and 0mC for q(0) in equation (1) to obtain the charge flowing through the device as follows:

q(t)=[(2t+5)mA]dt+0mC=[(2t)mA]dt+[(5)mA]dt=2(t22)mC+5tmC=(t2+5t)mC

Conclusion:

Thus, the charge q(t) flowing through the device is (t2+5t)mC_.

(c)

Expert Solution
Check Mark
To determine

Calculate the charge q(t) flowing through a device when the current is i(t)=20cos(10t+π/6)μA, and q(0)=2μC.

Answer to Problem 3P

The charge q(t) flowing through the device is 2[sin(10t+π/6)+1]μC_.

Explanation of Solution

Given data:

The current flowing through the device [i(t)] is 20cos(10t+π/6)μA.

The charge at initial state [q(0)] is 2μC.

Calculation:

Substitute 20cos(10t+π/6)μA for i(t) and 2μC for q(0) in equation (1) to obtain the charge flowing through the device as follows:

q(t)=[20cos(10t+π/6)μA]dt+2μC=20sin(10t+π/6)10μC+2μC {cos(at)dt=sin(at)a}=2sin(10t+π/6)μC+2μC=2[sin(10t+π/6)+1]μC

Conclusion:

Thus, the charge q(t) flowing through the device is 2[sin(10t+π/6)+1]μC_.

(d)

Expert Solution
Check Mark
To determine

Calculate the charge q(t) flowing through a device when the current is i(t)=10e30tsin(40t)A, and q(0)=0C.

Answer to Problem 3P

The cha charge q(t) flowing through the device is e30t[0.12sin(40t)+0.16cos(40t)]C_.

Explanation of Solution

Given data:

The current flowing through the device [i(t)] is 10e30tsin(40t)A.

The charge at initial state [q(0)] is 0C.

Calculation:

Substitute 10e30tsin(40t)A for i(t) and 0 C for q(0) in equation (1) to obtain the charge flowing through the device as follows:

q(t)=[10e30tsin(40t)A]dt+0C=10{[e30tsin(40t)]dt}C=10{e30t[(30)sin(40t)40cos(40t)](30)2+402+C}C {eatsin(bt)dt=eat[asin(bt)bcos(bt)]a2+b2+C}=10{e30t[30sin(40t)40cos(40t)]900+1600}C

Simplify the expression as follows:

q(t)=e30t[300sin(40t)2500400cos(40t)2500]C=e30t[0.12sin(40t)+0.16cos(40t)]C

Conclusion:

Thus, the charge q(t) flowing through the device is e30t[0.12sin(40t)+0.16cos(40t)]C_.

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Chapter 1 Solutions

Fundamentals of Electric Circuits

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