Question

In the case of an inductor,

• A. voltage lags the current by \( \frac{\pi}{2} \)
• B. voltage leads the current by \( \frac{\pi}{2} \)
• C. voltage leads the current by \( \frac{\pi}{3} \)
• D. voltage leads the current by \( \frac{\pi}{4} \)

Hint:

In an inductive circuit, the voltage leads the current by \( \frac{\pi}{2} \) radians due to the nature of inductance, which causes the voltage to react to changes in current.

Answer 1

The Correct Option is B

In an AC circuit containing an inductor, the relationship between voltage and current is characterized by their phase difference. For an ideal inductor, the voltage (\( V \)) across it can be represented in terms of the current (\( I \)), using the inductive reactance and the phase shift caused by the inductance. This phase difference is given by:

\( V(t) = L \frac{dI(t)}{dt} \)

The sinusoidal expressions for voltage and current in an AC circuit are:

\( I(t) = I_0 \sin(\omega t + \phi_I) \)

\( V(t) = V_0 \sin(\omega t + \phi_V) \)

For an inductor, voltage leads the current by \(\frac{\pi}{2}\) radians (or 90 degrees), meaning the voltage reaches its maximum value one-quarter of a cycle before the current reaches its maximum. Therefore, the phase difference (\( \phi_V - \phi_I \)) is:

\( \phi_V - \phi_I = \frac{\pi}{2} \)

Therefore, in an inductor, voltage leads the current by \( \frac{\pi}{2} \).

Answer 2

In an inductive circuit, the voltage \( V \) and the current \( I \) are related through the inductance \( L \) and the frequency of the alternating current. The voltage across an inductor is given by: \[ V_L = L \frac{dI}{dt} \] 
For an AC circuit, the current and voltage can be expressed as: \[ I = I_0 \sin(\omega t) \] \[ V_L = L I_0 \omega \cos(\omega t) \] Since \( \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \), the voltage leads the current by \( \frac{\pi}{2} \) radians in an inductor. 
Thus, the correct answer is: \[ \text{(B) } \text{voltage leads the current by } \frac{\pi}{2} \]