Explanation Formula used: The governing equation of Kirchhoff’s law is calculated as: L q ″ + R q ′ + 1 C q = E ( t ) . The quadratic equation a x 2 + b x + c = 0 is calculated as x = − b ± b 2 − 4 a c 2 a . Theorem used: “If r 1 = α + i β and r 1 = α − i β are the two complex roots of the auxiliary equation a r 2 + b r + c = 0 , then y = e α x ( c 1 cos β x + c 2 sin β x ) is the general solution of a y ″ + b y ′ + c y = 0 . The general solution y = y ( x ) to the nonhomogeneous differential equation a y ″ + b y ′ + c y = G ( x ) has the form y = y c + y p , where the complementary solution y c is the general solution to the associated homogeneous equation a y ″ + b y ′ + c y = 0 , and y p is any particular solution to the nonhomogeneous equation a y ″ + b y ′ + c y = G ( x ) ”. Calculation: The voltage source is: E ( t ) = 0 The resistance R is 8.00 Ω , the capacitance C is 1.00 μ F , and the inductance L is 0.200 H . Now, the governing equation of Kirchhoff’s law is: L q ″ + R q ′ + 1 C q = E ( t ) Where, q is the charge at a cross section of a conductor, measured in coulombs, I is the current or rate of change d q d t (flow of electrons) at a cross section of a conductor, measured in amperes, E is the electric (potential) source, measured in volts, and V is the difference in potential between two points along the conductor, measured in volts ( v ). Substitute the resistance R is 8.00 Ω , the capacitance C is 1.00 μ F , and the inductance L is 0.200 H . in L q ″ + R q ′ + 1 C q = E ( t ) . 0.200 d 2 q d t 2 + 8.00 d q d t + 10 6 q = 0 The auxiliary equation is: 0.200 m 2 + 8.00 m + 10 6 = 0 Compare the above equation with a x 2 + b x + c = 0 . Here a = 0.200 , b = 8.00 , c = 10 6 . Substitute the value of a = 0.200 , b = 8.00 , c = 10 6 in the quadratic formula x = − b ± b 2 − 4 a c 2 a . m = − 8 ± 8 2 − 4 × 0.2 × 10 6 2 × 0.2 = − 8 ± 64 − 0.8 × 10 6 0.4 = − 20 ± 2240 j So, the general solution of differential equation is: q = e − 20 t ( c 1 cos 2240 t + c 2 sin 2240 t ) Substitute the boundary condition q = 0 coulombs when t = 0 in q = e − 20 t ( c 1 cos 2240 t + c 2 sin 2240 t )

11th Edition

ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 31.10, Problem 17E

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EBK BASIC TECHNICAL MATHEMATICS

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

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