Question

A given AC source, v = v₀ sin ωt,  is connected across a given series LCR circuit. The graph, showing simultaneously, the variation of the voltage and current in the circuit, with time, has the form shown. The current, flowing in this circuit, has to be made to have its maximum value (=v/R). This aim can be achieved if, without making any other change in the circuit, we join another inductor (L"), of value

(1) \((\frac{1-ω^2 LC}{ω^2C})\) in series with the given inductor (L)

(2) \((\frac{1-ω^2 LC}{ω^2C})\) in parallel with the given inductor (L)

(3) \((\frac{ω^2 LC}{ω^2C})\)in series with the given inductor (L)

(4) \((\frac{ω^2 C}{1-ω^2LC})\)in series with the given inductor (L)

Answer 1

(1) \((\frac{1-ω^2 LC}{ω^2C})\) in series with the given inductor (L)

The given graph shows the current leading in phase with respect to the applied voltage. The given condition (current has its maximum value (= V/K) is met when the circuit is in its resonance state. For this, the net inductance, has to have a value, L’, such that ωL' = 1/ωC. This requires L' = 1/ω²C.

Since the current is leading in phase, we need to increase, the value of the inductance. We, therefore, need to add another inductor (of value L”) in series with the given inductor, such that