In a damped LCR circuit, the time required for the charge amplitude to decrease to half of its initial value is t₁ . The time required for the maximum total electrical energy of the circuit to decrease to half of its initial value is t₂ . T

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In a damped LCR circuit, the time required for the charge amplitude to decrease to half of its initial value is t₁ . The time required for the maximum total electrical energy of the circuit to decrease to half of its initial value is t₂ . The ratio t₁ : t₂ is:

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Correct answer: C. 2 : 1

The charge amplitude in a damped LCR circuit decays exponentially as: Q = Q₀ e^ - Rt 2L For the amplitude to drop to half ( Q = Q₀ 2 ): 1 2 = e^ - R t₁ 2L t₁ = 2L R (2) The total electrical energy U is proportional to the square of the charge amplitude ( U Q^2 ), so it decays as: U = U₀ e^ - Rt L For the energy to drop to half ( U = U₀ 2 ): 1 2 = e^ - R t₂ L t₂ = L R (2) Taking the ratio of the two times: t₁ t₂ = 2L R (2) L R (2) = 2 Thus, the ratio t₁ : t₂ is 2 : 1 . Answer: 2 : 1

In a damped LCR circuit, the time required for the charge amplitude to decrease to half of its initial value is t₁ . The time required for the maximum total electrical energy of the circuit to decrease to half of its initial value is t₂ . The ratio t₁ : t₂ is:

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