Two series LCR circuits, Circuit 1 and Circuit 2, are connected to the same variable frequency AC source of peak voltage V₀ . Circuit 1 has inductance L , capacitance C , and resistance R . Circuit 2 has inductance 2L , capacitance C 2 , and resistance 2R . If f₁ and f₂ are their respective resonant frequencies, and I₁ and I₂ are their respective maximum peak currents at resonance, which of the following relationship

Step-by-step solution

The resonant frequency of a series LCR circuit is given by f = 1 2 LC . For Circuit 1: f₁ = 1 2 LC For Circuit 2: f₂ = 1 2 (2L) ( C 2 ) = 1 2 LC Thus, f₁ = f₂ . At resonance, the impedance of the circuit is minimum and equal to its resistance ( Z = R ). The maximum peak current is I = V₀ R . For Circuit 1: I₁ = V₀ R For Circuit 2: I₂ = V₀ 2R = 1 2 ( V₀ R ) = I₁ 2 Thus, I₁ = 2I₂ . Answer: f₁ = f₂ and I₁ = 2I₂