In a pure L-C series circuit, the value of phase differences between voltages across the inductor and the capacitor will be ________.

Explanation:

Correct Option Analysis:

The correct option is:

Option 3: 180°

This option accurately describes the phase difference between voltages across the inductor and the capacitor in a pure L-C series circuit. To understand why this is the case, let's delve into the fundamental principles of AC circuits involving inductors and capacitors.

Understanding Phase Difference in L-C Series Circuit:

In an L-C series circuit, the inductor (L) and capacitor (C) are connected in series with each other and are subjected to an alternating current (AC) source. The behavior of the voltages across these components is fundamentally different due to their distinct reactive properties:

• Inductor (L): An inductor opposes changes in current and creates a voltage that leads the current by 90 degrees. This means the voltage across the inductor reaches its maximum value one-quarter cycle before the current does.
• Capacitor (C): A capacitor, on the other hand, opposes changes in voltage and creates a voltage that lags the current by 90 degrees. This means the voltage across the capacitor reaches its maximum value one-quarter cycle after the current does.

Because of these properties, the voltage across the inductor (V_L) and the voltage across the capacitor (V_C) are 180 degrees out of phase. This 180-degree phase difference is a direct result of the fact that the voltage across the inductor leads the current by 90 degrees, while the voltage across the capacitor lags the current by 90 degrees. Therefore, the voltage across the inductor and the voltage across the capacitor are in complete opposition to each other.

Mathematical Representation:

Let's consider an AC current I(t) = I₀sin(ωt) flowing through the L-C series circuit:

• Voltage across the Inductor (V_L): V_L(t) = L(dI/dt) = LωI₀cos(ωt) = V_L0cos(ωt)
• Voltage across the Capacitor (V_C): V_C(t) = (1/C)∫I(t)dt = (I₀/Cω)cos(ωt) = V_C0cos(ωt)

Here, ω is the angular frequency, and the voltages V_L(t) and V_C(t) are 180 degrees out of phase, leading to V_L(t) = -V_C(t).

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 30°

This option is incorrect because the phase difference between the voltages across an inductor and a capacitor in a pure L-C series circuit is not 30 degrees. As explained earlier, the phase difference is a result of the inductive and capacitive reactances, which cause the voltages to be 180 degrees out of phase.

Option 2: zero

This option is also incorrect. A zero-degree phase difference would imply that the voltages across the inductor and capacitor are in phase with each other, which contradicts their reactive properties. The inductive voltage leads the current by 90 degrees, while the capacitive voltage lags the current by 90 degrees, resulting in a total phase difference of 180 degrees.

Option 4: 90°

This option is incorrect as well. A 90-degree phase difference is the phase difference between the voltage and current for either an inductor or a capacitor individually. However, in the context of an L-C series circuit, the voltages across the inductor and capacitor themselves are 180 degrees out of phase.

Conclusion:

Understanding the phase differences in an L-C series circuit is crucial for analyzing AC circuits. In a pure L-C series circuit, the voltage across the inductor and the voltage across the capacitor are 180 degrees out of phase due to their respective leading and lagging characteristics with respect to the current. This phase opposition is fundamental to the behavior of reactive components in AC circuits, highlighting the importance of their unique properties in electrical engineering.