A circuit containing resistance R ₁ , inductance L ₁ and capacitance C ₁ connected in series resonates at the same frequency ' f _ r ' as another circuit containing R ₂, ~L ₂ and C ₂ in series. If two circuits are connected in series, then the new frequency at resonance is

Step-by-step solution

When R ₁, ~L ₁ and C ₁ are connected in series the resonant (angular) frequency is given by _ r = 1 L ₁ C ₁ Also, when R ₂, ~L ₂, C ₂ are connected in series Then _ r = 1 L ₂ C ₂ By (1) and (2): L ₁ C ₁= L ₂ C ₂ If the two circuits are connected in series then the equivalent inductance is given by L = L ₁+ L ₂ and the equivalent capacitance is given by aligned & C = C ₁ C ₂ C ₁+ C ₂ & LC = ( L ₁+ L ₂ ) C ₁ C ₂ C ₁+ C ₂ & = L ₁ C ₁ C ₂+ L ₂ C ₁ C ₂ C ₁+ C ₂ [ L ₁ C ₁= L ₂ C ₂ ] & = L ₂ C ₂ ( C ₂+ C ₁ ) C ₁+ C ₂ = L