Phase Relations In Ac Elements

For a pure capacitor the instantaneous charge is (q = CV), and the current is the rate of change of charge, (i = C,dV/dt). Differentiating a sinusoidal voltage (V = V_0\sin\omega t) gives (i = \omega C V_0\cos\omega t = \omega C V_0\sin(\omega t + 90^\circ)), showing the current reaches its peak a quarter cycle before the voltage. Hence the current leads the applied voltage by ninety degrees. The option 'lags by 90 degrees' describes a pure inductor, where current builds up only after voltage is applied. The 'in phase' option applies to a pure resistor, not a capacitor. The '45 degrees' option would require a mix of resistance and reactance, not a single ideal capacitor. This quadrature relationship is exactly the NCERT result underlying why a capacitor dissipates no average power, because the average of the product of a sine and a cosine over a complete cycle is zero. The same reasoning shows that the magnitude of the current is limited by the capacitive reactance (X_C = 1/(\omega C)), so larger capacitance or higher frequency permits a bigger current. A consistency check notes that since current peaks while voltage is zero and rising fastest, the energy merely oscillates into and out of the electric field rather than being consumed, confirming the leading-current, lossless behaviour of ideal capacitive elements.