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

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

8th Edition

ISBN: 9781305779198

Author: James Stewart

Publisher: Cengage Learning

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Chapter 17.3, Problem 13E

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

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Chapter 17.3, Problem 12E

Chapter 17.3, Problem 14E

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Chapter 17 Solutions

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

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...

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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.

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

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Chapter 17 Solutions