In a damped oscillating LCR circuit, the amplitude of the oscillating charge at t=0 is Q₀ . If the charge amplitude at time t₀ is Q₁ and at time 2t₀ is Q₂ , which of the following algebraic relationships is correct?

Step-by-step solution

In a damped LCR circuit, the amplitude of the oscillating charge decays exponentially with time according to the relation: Q(t) = Q₀ e^ - Rt 2L At time t = t₀ , the amplitude is: Q₁ = Q₀ e^ - Rt₀ 2L e^ - Rt₀ 2L = Q₁ Q₀ At time t = 2t₀ , the amplitude is: Q₂ = Q₀ e^ - R(2t₀) 2L = Q₀ (e^ - Rt₀ 2L )^2 Substituting the value of the exponential term: Q₂ = Q₀ ( Q₁ Q₀ )^2 = Q₁^2 Q₀ Rearranging the equation, we get: Q₁^2 = Q₀ Q₂ This shows that the charge amplitudes at equally spaced time intervals form a geometric progres