Explanation 1) Concept: The damped oscillations in RLC (resistance, inductor and capacitor) circuit contain electromagnetic energy. Damped oscillations means oscillations, which fade away with time. In this problem, charge on the capacitor is given. After N complete cycles of oscillations, the charge on the capacitor will vary. To calculate the change in the charge on capacitor, we have, q = Q e - R t / 2 L c o s ⁡ ( ω ' t + φ ) . As given in the problem, t changes with the number of cycles. Therefore, t = NT = 2 π N / ω ' as we know t = 2 π / ω . 2) Given: i) Resistance, R = 7.2 Ω ii) Inductance, L = 12 H iii) Capacitance, C = 3.20 μ F iv) Capacitor charge, q 0 = 6.2 μ C 3) Formula: i) q = Q e - R t / 2 L c o s ⁡ ( ω ' t + φ ) ii) t = 2 π / ω 4) Calculation: To calculate the change in the charge capacitor due to number of oscillatory cycles, first we will consider an equation for damped oscillation charge capacitor: q = Q e - R t / 2 L c o s ⁡ ( ω ' t + φ ) The charge q after N cycles is obtained by substituting t = NT = 2 π N / ω ' into the above equation: q = Q e - R N T / 2 L c o s ⁡ ( ω ' ( 2 π N / ω ' ) + φ ) q = Q e - R N ( 2 π L C / 2 L c o s ⁡ ( 2 π N + φ ) where, ω = L C q = Q e - R N ( 2 π L C / 2 L L ) c o s ⁡ ( 2 π N + φ ) q = Q e - N π R C / L cos ⁡ φ … … … … … … …

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Chapter 31, Problem 24P

To determine

To find

a) Charge on capacitor for cycle N = 5

b) Charge on capacitor for cycle N = 10

c) Charge on capacitor for cycle N = 100

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Chapter 31, Problem 23P

Chapter 31, Problem 25P

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To find a) Charge on capacitor for cycle N = 5 b) Charge on capacitor for cycle N = 10 c) Charge on capacitor for cycle N = 100

Solution Summary: The author explains that the damped oscillations in RLC (resistance, inductor and capacitor) circuit contain electromagnetic energy.

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To find a) Charge on capacitor for cycle N = 5 b) Charge on capacitor for cycle N = 10 c) Charge on capacitor for cycle N = 100

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Chapter 31 Solutions

To find

a) Charge on capacitor for cycle N = 5

b) Charge on capacitor for cycle N = 10

c) Charge on capacitor for cycle N = 100