Explanation Given data: Series RLC circuit with following values. R = 20 Ω , L = 1 H , C = 0 . 0 0 2 F , E ( t ) = 12 V , Q ( 0 ) = 0 and Q ′ ( 0 ) = I ( 0 ) = 0 . Formula used: Write the expression for differential equation of electric circuit. L d 2 Q d t 2 + R d Q d t + 1 C Q = E ( t ) L Q ″ + R Q ′ + 1 C Q = E ( t ) (1) Write the expression for general solution with complex roots. Q ( t ) = e α t [ c 1 cos ( β t ) + c 2 sin ( β t ) ] (2) Write the expression for r . r = α + β i (3) Write the expression for auxiliary equation. a r 2 + b r + c = 0 (4) Write the expression for differential equation. a y ″ + b y ′ + c y = 0 (5) Find the differential equation for electric circuit using equation (1). Substitute 1 for L , 20 for R , 0 . 0 0 2 for C and 12 for E ( t ) in equation (1), 1 Q ″ + 20 Q ′ + 1 0.002 Q = 12 Q ″ + 20 Q ′ + 500 Q = 12 (6) Modify equation (5) as follows. a Q ″ + b Q ′ + c Q = 0 (7) Compare equation (6) and (7). a = 1 b = 20 c = 500 Substitute 1 for a , 20 for b and 500 for c in equation (4), r 2 + 20 r + 500 = 0 Find the value of r . r = − 20 ± ( 20 ) 2 − 4 ( 1 ) ( 500 ) 2 ( 1 ) = − 20 ± 400 − 2000 2 = − 20 ± − 1600 2 = − 20 ± 40 i 2 Simplify r as follows. r = − 10 ± 20 i (8) Compare equations (3) and (8). α = − 10 β = 20 Substitute − 10 for α and 20 for β in equation (2), Q ( t ) = e − 10 t [ c 1 cos ( 20 t ) + c 2 sin ( 20 t ) ] (9) Modify equation (9) for complementary equation as follows. Q c ( t ) = e − 10 t [ c 1 cos ( 20 t ) + c 2 sin ( 20 t ) ] Consider the value of Q p ( t ) as follows. Q p ( t ) = A (10) Differentiate equation (10) with respect to t . Q ′ p ( t ) = 0 Differentiate equation with respect to t . Q ″ p ( t ) = 0 Modify equation (6) as follows. Q ″ p ( t ) + 20 Q ′ p ( t ) + 500 Q p ( t ) = 12 Substitute 0 for Q ″ p ( t ) , 0 for Q ′ p ( t ) and A for Q p ( t ) , 1 ( 0 ) + 20 ( 0 ) + 500 ( A ) = 12 500 ( A ) = 12 A = 12 500 A = 3 125 Substitute 3 125 for A in equation (10), Q p ( t ) = 3 125 Write the expression for general solution. Q ( t ) = Q c ( t ) + Q p ( t ) Substitute e − 10 t [ c 1 cos ( 20 t ) + c 2 sin ( 20 t ) ] for Q c ( t ) and 3 125 for Q p ( t ) , Q ( t ) = e − 10 t [ c 1 cos ( 20 t ) + c 2 sin ( 20 t ) ] + 3 125 (11) Substitute 0 for t , Q ( 0 ) = e − 10 ( 0 ) [ c 1 cos ( 20 × 0 ) + c 2 sin ( 20 × 0 ) ] + 3 125 = c 1 + 3 125 Substitute 0 for Q ( 0 ) , 0 = c 1 + 3 125 c 1 = 0 − 3 125 c 1 = − 3 125 Write the expression for I

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 17.3, Problem 13E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.3, Problem 12E

Chapter 17.3, Problem 14E

Students have asked these similar questions

Find the derivative of the function. 5 1 6 p(x) = -24x 5 +15x

∞ 2n (4n)! Let R be the radius of convergence of the series -x2n. Then the value of (3" (2n)!)² n=1 sin(2R+4/R) is -0.892 0.075 0.732 -0.812 -0.519 -0.107 -0.564 0.588

Find the cost function if the marginal cost function is given by C'(x) = x C(x) = 2/5 + 5 and 32 units cost $261.

Chapter 17 Solutions

Multivariable Calculus

Ch. 17.1 - Solve the differential equation. 1. y" y' 6y = 0 Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Solve the differential equation. 4. y" + y' 12y =...Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Solve the differential equation. 9. y" 4y' + 13y...Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Solve the initial-value problem. 19. 9y" + 12y' +...Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Solve the initial-value problem. 23. y" y' 12y =...Ch. 17.1 - Solve the initial-value problem. 24. 4y" + 4y' +...Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Solve the boundary-value problem, if possible. 32....Ch. 17.1 - Prob. 33E Ch. 17.1 - If a, b, and c are all positive constants and y(x)...Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.3 - A spring has natural length 0.75 m and a 5-kg...Ch. 17.3 - Prob. 2E Ch. 17.3 - A spring with a mass of 2 kg has damping constant...Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Suppose a spring has mass m and spring constant k...Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - The battery in Exercise 14 is replaced by a...Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - Prob. 2RCC Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - Prob. 4RCC Ch. 17 - Prob. 5RCC Ch. 17 - Prob. 1RQ Ch. 17 - Prob. 2RQ Ch. 17 - Prob. 3RQ Ch. 17 - Prob. 4RQ Ch. 17 - Solve the differential equation. 1. 4y" y =0 Ch. 17 - Prob. 2RE Ch. 17 - Prob. 3RE Ch. 17 - Solve the differential equation. 4. y" + 8y' + 16y...Ch. 17 - Prob. 5RE Ch. 17 - Prob. 6RE Ch. 17 - Prob. 7RE Ch. 17 - Prob. 8RE Ch. 17 - Prob. 9RE Ch. 17 - Solve the differential equation. 10....Ch. 17 - Prob. 11RE Ch. 17 - Solve the initial-value problem. 12. y" 6y' + 25y...Ch. 17 - Solve the initial-value problem. 13. y" 5y' + 4y...Ch. 17 - Prob. 14RE Ch. 17 - Solve the boundary-value problem, if possible. 15....Ch. 17 - Prob. 16RE Ch. 17 - Prob. 17RE Ch. 17 - Prob. 18RE Ch. 17 - A series circuit contains a resistor with R = 40 ,...Ch. 17 - Prob. 20RE Ch. 17 - Assume that the earth is a solid sphere of uniform...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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

Solution Summary: The author explains the charge and current of series RLC circuit at time t. The formula used: Write the expression for differential equation of electric circuit.

Videos

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

Want to see the full answer?

Chapter 17 Solutions