Match List-I with List-II.
List-I
A. Phase difference between current and voltage in a purely resist

Q: Why does current lag in an inductor?

An inductor opposes any change in current by inducing a back EMF. This opposition delays the rise of current, causing it to lag behind the applied voltage.

Q: Why does current lead in a capacitor?

A capacitor needs current to flow into it before a voltage can build up across its plates. Therefore, the current must peak before the voltage does, meaning current leads voltage.

Q: What happens if $X_L = X_C$ in an L-C-R circuit?

This is the condition for resonance. The net reactance becomes zero, and the circuit behaves like a purely resistive circuit with a phase difference of zero.

Q: Is the formula for phase difference always $\tan^{-1}(\frac{X_C - X_L}{R})$?

It depends on the convention. If you define $\phi$ as the angle by which current leads voltage, then it is $\frac{X_C - X_L}{R}$. If you define it as the angle by which voltage leads current, it is $\frac{X_L - X_C}{R}$.

Q: Can the phase difference in an L-C-R circuit be exactly $90^\circ$?

Only if the resistance $R$ is exactly zero, which makes it a purely LC circuit. In any real L-C-R circuit with some resistance, the phase difference will be strictly between $-90^\circ$ and $+90^\circ$.

Question

Match List-I with List-II.

\begin{array}{|l|l|}
\hline
	ext{List-I} & 	ext{List-II} \
\hline
	ext{A. Phase difference between current and voltage in a purely resistive AC circuit} & 	ext{1. } \frac{\pi}{2}	ext{; current leads voltage} \
	ext{B. Phase difference between current and voltage in a pure inductive AC circuit} & 	ext{2. zero} \
	ext{C. Phase difference between current and voltage in a pure capacitive AC circuit} & 	ext{3. } \frac{\pi}{2}	ext{; current lags voltage} \
	ext{D. Phase difference between current and voltage in an L-C-R series circuit} & 	ext{4. } 	an^{-1}\left(\frac{X_C - X_L}{R}ight) \
\hline
\end{array}

Choose the most appropriate answer from the options given below.

Options:

(A) A $ightarrow$ 1, B $ightarrow$ 3, C $ightarrow$ 4, D $ightarrow$ 2

(B) A $ightarrow$ 2, B $ightarrow$ 4, C $ightarrow$ 3, D $ightarrow$ 1

(C) A $ightarrow$ 2, B $ightarrow$ 3, C $ightarrow$ 4, D $ightarrow$ 1

(D) A $ightarrow$ 2, B $ightarrow$ 3, C $ightarrow$ 1, D $ightarrow$ 4

Sigma Percentile · Accepted Answer

### Analyzing the Setup
In alternating current (AC) circuits, the relationship between voltage and current is defined by the phase difference. This phase difference arises because different components—resistors, inductors, and capacitors—respond differently to changing currents and voltages. Let's break down each component to understand how it affects the phase.

### Purely Resistive Circuit
Imagine a circuit with only a resistor. A resistor simply opposes the flow of current without storing any energy. Because of this, the current and voltage rise and fall together in perfect harmony. They are completely in phase.

Therefore, the phase difference $\phi$ is exactly $0$. This matches List-I entry A with List-II entry 2.

### Purely Inductive Circuit
Now, consider a purely inductive circuit. An inductor is like a heavy flywheel; it opposes any change in the current flowing through it due to self-induction. When an AC voltage is applied, the inductor fights the rising current, causing the current to lag behind the voltage.

Mathematically, the current lags the voltage by exactly $90^\circ$ or $\frac{\pi}{2}$ radians. This matches List-I entry B with List-II entry 3.

### Purely Capacitive Circuit
Next, let's look at a purely capacitive circuit. A capacitor stores electrical energy. When an AC voltage is applied, the current must flow first to build up the charge on the capacitor's plates before the voltage across it can rise.

Because the current acts first, the current leads the voltage by exactly $90^\circ$ or $\frac{\pi}{2}$ radians. This matches List-I entry C with List-II entry 1.

### The L-C-R Series Circuit
Finally, what happens when we put all three components together in series? The overall phase difference depends on the competition between the inductive reactance $X_L$ and the capacitive reactance $X_C$.

The net reactance is $X_C - X_L$ (or vice versa), and the phase angle $\phi$ is determined by the ratio of this net reactance to the resistance $R$.

The formula is given by:
$$ 	an \phi = \frac{X_C - X_L}{R} $$

Taking the inverse tangent, we get:
$$ \phi = 	an^{-1}\left(\frac{X_C - X_L}{R}ight) $$

This matches List-I entry D with List-II entry 4.

### Final Conclusion
Putting it all together, we have:
- **A $ightarrow$ 2**
- **B $ightarrow$ 3**
- **C $ightarrow$ 1**
- **D $ightarrow$ 4**

This corresponds perfectly to option (d).

Select Answer:

Visualized Solution

$R Circuit$

$L Circuit$

$C Circuit$

$L - C - R Circuit$

$Final Match$

The Sigma Insight: AC Circuits and Power in AC Circuits

Analyzing the Setup In alternating current (AC) circuits, the relationship between voltage and current is defined by the phase difference

Purely Resistive Circuit Imagine a circuit with only a resistor

Purely Inductive Circuit Now, consider a purely inductive circuit

Purely Capacitive Circuit Next, let's look at a purely capacitive circuit

The L-C-R Series Circuit

Final Conclusion

Similar Questions

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L−C−R Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Match