Question

An ac source of voltage \( V = V_0 \sin \omega t \) is connected to a circuit element \( X \). It is observed that the current flowing through \( X \) varies as \( I = I_0 \sin\left(\omega t - \frac{\pi}{2}\right) \).
(a) Identify the element \( X \) and write the expression for its reactance.

Hint:

In AC circuits: - If current lags voltage by \( \frac{\pi}{2} \), the component is an inductor. - If current leads voltage by \( \frac{\pi}{2} \), the component is a capacitor. - Reactance of inductor: \( X_L = \omega L \), increases with frequency.

Answer 1

In an AC circuit, when the current lags behind the voltage by \( \frac{\pi}{2} \), the element causing this phase difference is a pure inductor. This is because in an inductor, voltage leads current by 90°, or equivalently, current lags voltage by 90°. Hence, the element \( X \) is an inductor. The reactance of an inductor is given by: \[ X_L = \omega L \] where \( X_L \) = inductive reactance, \( \omega \) = angular frequency of AC supply, \( L \) = inductance of the coil.