Explanation Given: The total energy stored in the capacity ( W ) Capacitor of Capacitance ( C ) Voltage source ( V ) Power extended by time t is P ( t ) The resistance in the circuit ( R ) P ( t ) = V 2 R ( e − t / R C − e − 2 t / R C ) W = ∫ 0 ∞ P ( t ) ⋅ d t Key concepts applied: The total energy contained after the capacitor is fully charge is obtained by substituting the function P ( t ) in the integral W = ∫ 0 ∞ P ( t ) ⋅ d t to get W = V 2 R ∫ 0 ∞ ( e − t / R C − e − 2 t / R C ) ⋅ d t The energy after a finite amount of time ( t = T ) is V 2 R ∫ 0 T ( e − t / R C − e − 2 t / R C ) ⋅ d t Calculation: W = V 2 R ∫ 0 T ( e − t / R C − e − 2 t / R C ) ⋅ d t = V 2 R [ e − t / R C − 1 R C − e − 2 t / R C − 2 R C ] 0 T = V 2 R [ − R C ⋅ e − t / R C − R C e − 2 t / R C − 2 ] 0 T = V 2 R [ − R C ⋅ e − t / R C + R C e − 2 t / R C 2 ] 0 T = V 2 R

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.7, Problem 83E

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The total energy stored in the capacity ( $W$ ) is half of product of capacitance ( $C$ ) with square of voltage source ( $V$ )

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Chapter 8.7, Problem 82E

Chapter 8.7, Problem 84E

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Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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The total energy stored in the capacity ( W ) is half of product of capacitance ( C ) with square of voltage source ( V )

Solution Summary: The author explains that the total energy stored in the capacity is half of product of capacitance ( C ) with square of voltage source ( V ).

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To Show:

The total energy stored in the capacity ( $W$ ) is half of product of capacitance ( $C$ ) with square of voltage source ( $V$ )

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Chapter 8 Solutions

The total energy stored in the capacity ( W ) is half of product of capacitance ( C ) with square of voltage source ( V )

Solution Summary: The author explains that the total energy stored in the capacity is half of product of capacitance ( C ) with square of voltage source ( V ).

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To Show: The total energy stored in the capacity (W) is half of product of capacitance (C) with square of voltage source (V)

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Chapter 8 Solutions

To Show:

The total energy stored in the capacity ( $W$ ) is half of product of capacitance ( $C$ ) with square of voltage source ( $V$ )