In an a.c. circuit with pure capacitance 'C' and a.c. source $E = E_0\sin\omega t$, the equation of instantaneous current is given by

In a series LCR circuit alternating e.m.f. and current are given by the equations $V = V_0\sin(\omega t)$ and $I = I_0\sin\left(\omega t + \dfrac{\pi}{3}\right)$ respectively. The average power dissipated in the circuit over one cycle of a.c. is $(\cos 60^\circ = 0.5)$

The resistor '$R$' inductance '$L$' and capacitance '$C$' are connected in series with an a.c. source. When 'L' is removed from the circuit, the phase difference between voltage and current in the circuit is $\dfrac{\pi}{3}$. If instead, C is removed from the circuit, the phase difference is again $\dfrac{\pi}{3}$. The power factor of the circuit is

The LC series resonant circuit produces a resonant frequency '$f$'. If L is tripled and 'C' is increased by 3C, the resonant frequency will be