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In the given circuit the capacitor \( (C) \) may be charged through resistance \( R \) by a battery \( V \) by closing switch \( S_{1} \). Also when \( S_{1} \) is opened and \( S_{2} \) is closed the capacitor is connected in series with inductor \( (L) \).

At the start, the capacitor was uncharged. When switch \( S_{1} \) is closed and \( S_{2} \) is kept open, the time constant of this circuit is \( \tau \). Which of the following is correct?
(1) after time interval \( \tau \), charge on the capacitor is
\[
\frac{\mathrm{CV}}{2}
\]
(2) after time interval \( 2 \tau \), charge on the capacitor is \( C V\left(1-e^{-2}\right) \)
(3) the work done by the voltage source will be half of the heat dissipated within the capacitor is fully charged
(4) after time interval \( 2 \tau \), charge on the capacitor is \( C V\left(1-e^{-1}\right) \)


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