First slide

AC Circuits

Question

In an LC circuit shown in figure, C = 1F, L = 4H. At time t = 0, charge in the capacitor is 4C and it is decreasing at the rate of $\sqrt{5} {Cs}^{- 1}$ . Choose the correct statement.

Difficult

A

Maximum charge in the capacitor can be 6 C
B

Maximum charge in the capacitor can be 8 C
C

Charge in the capacitor will be maximum after time 3 sin^-1 (2/3) s
D

None of these

Solution

$\begin{array}{l} G i v e n : \frac{d q}{d t} = i = \sqrt{5} A \\ A t s o m e t i m e t, \\ \Rightarrow \frac{q_{m}^{2}}{2 C} = \frac{q^{2}}{2 C} + \frac{1}{2} L i^{2} \\ \Rightarrow \frac{q_{m}^{2}}{2} = \frac{16}{2} + \frac{1}{2} (4) (5) = 18 \\ \Rightarrow q_{max} = 6 C \end{array}$
$q = Q \cos (ω t + δ) a t t = 0, q = 4$
$\Rightarrow 4 = 6 \cos δ \Rightarrow \cos δ = \frac{2}{3} \Rightarrow δ = \cos^{- 1} (\frac{2}{3})$
Charge on the capacitor is maximum at t = t₁for this $ω t_{1} + δ = π$
$\Rightarrow t_{1} = \frac{1}{ω} (π - δ) = \sqrt{L C} (π - δ) = 2 [π - \cos^{- 1} (\frac{2}{3})]$

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