Understanding AC Circuits and Phase Difference

In alternating current (AC) circuits, the voltage (electromotive force or emf) and the current can be in phase, or one can lead or lag the other. This phase difference depends on the components present in the circuit, such as resistors, inductors, and capacitors.

Let's examine the standard phase relationships for the types of AC circuits mentioned in List - II:

• (I) A.C. circuit containing only an inductor: In a purely inductive AC circuit, the emf leads the current by a phase angle of $\frac{\pi}{2}$ radians (or 90 degrees). Equivalently, the current lags behind the emf by $\frac{\pi}{2}$.
• (III) A.C. circuit containing only a capacitor: In a purely capacitive AC circuit, the current leads the emf by a phase angle of $\frac{\pi}{2}$ radians (or 90 degrees). Equivalently, the emf lags behind the current by $\frac{\pi}{2}$.
• (II) A.C. circuit with resistance and inductance in series: In a series R-L AC circuit, the emf leads the current by a phase angle $\phi$, where $0 < \phi < \frac{\pi}{2}$. The exact value of $\phi$ depends on the ratio of inductive reactance ($X_L$) to resistance (R), given by $\tan(\phi) = \frac{X_L}{R}$. The current lags behind the emf by $\phi$.
• (IV) A.C. circuit with resistance and capacitor in series: In a series R-C AC circuit, the current leads the emf by a phase angle $\phi$, where $0 < \phi < \frac{\pi}{2}$. The exact value of $\phi$ depends on the ratio of capacitive reactance ($X_C$) to resistance (R), given by $\tan(\phi) = \frac{X_C}{R}$. The emf lags behind the current by $\phi$.

Now, let's match the descriptions in List - I with the circuits in List - II based on the provided correct mapping.

Analyzing the Matchings from the Provided Answer

Let's look at each pairing indicated by the provided correct answer:

• (A) $\leftrightarrow$ (III): List - I (A) states "The current lags behind the emf by $\frac{\pi}{4}$ in phase". List - II (III) is "A.C. circuit containing only a capacitor". In a pure capacitor circuit, the current leads the emf by $\frac{\pi}{2}$.
• (B) $\leftrightarrow$ (II): List - I (B) states "The e.m.f. lags behind the current by $\frac{\pi}{4}$ in phase". List - II (II) is "A.C. circuit with resistance and inductance in series". In a series R-L circuit, the emf leads the current by an angle between 0 and $\frac{\pi}{2}$, depending on R and $X_L$. This means the current lags the emf by that angle. If emf lags current by $\frac{\pi}{4}$, it means current leads emf by $\frac{\pi}{4}$, which happens in an R-C circuit with $R=X_C$.
• (C) $\leftrightarrow$ (IV): List - I (C) states "The current lags behind the e.m.f. in phase by $\frac{\pi}{2}$". List - II (IV) is "A.C. circuit with resistance and capacitor in series". In a series R-C circuit, the current leads the emf by an angle between 0 and $\frac{\pi}{2}$. Current lags emf by $\frac{\pi}{2}$ in a pure inductor circuit.
• (D) $\leftrightarrow$ (I): List - I (D) states "The e.m.f. lags behind the current in phase by $\frac{\pi}{2}$". List - II (I) is "A.C. circuit containing only an inductor". In a pure inductor circuit, the emf leads the current by $\frac{\pi}{2}$. Emf lags current by $\frac{\pi}{2}$ in a pure capacitor circuit.

Based on the provided correct answer's mapping, we have:

• (A) matches with (III)
• (B) matches with (II)
• (C) matches with (IV)
• (D) matches with (I)

This corresponds to the matching option (A)-(III), (B)-(II), (C)-(IV), (D)-(I).

Revision Table: Key AC Circuit Phase Relations

Additional Information on AC Circuit Components

Each component (resistor, inductor, capacitor) behaves differently when subjected to an alternating voltage. This different behavior is reflected in the relationship between the voltage across the component and the current flowing through it, specifically in their phase difference.

• Resistor: A resistor opposes the flow of current, but the voltage across it and the current through it change direction at the same time. They are in phase. The opposition to current is its resistance, R.
• Inductor: An inductor opposes changes in current by generating a back emf. This causes the voltage across the inductor to reach its peak value before the current reaches its peak value. The opposition to AC current is called inductive reactance ($X_L = \omega L$), where $\omega$ is the angular frequency and L is the inductance.
• Capacitor: A capacitor stores electrical energy in an electric field. As the AC voltage changes, the capacitor charges and discharges. The current flows into the capacitor to charge it before the voltage across it builds up. This causes the current to reach its peak value before the voltage across it. The opposition to AC current is called capacitive reactance ($X_C = \frac{1}{\omega C}$), where C is the capacitance.
• Series Circuits (R-L, R-C): When components are combined in series, the total opposition to current is called impedance (Z). The phase difference in these circuits depends on the relative values of resistance and reactance. For R-L, the impedance $Z = \sqrt{R^2 + X_L^2}$ and the phase angle $\phi = \arctan(\frac{X_L}{R})$. For R-C, the impedance $Z = \sqrt{R^2 + X_C^2}$ and the phase angle $\phi = \arctan(\frac{X_C}{R})$. The total voltage is the vector sum of the voltages across each component.