In an LCR series AC circuit, the voltage across each of the...

In an LCR series AC circuit, the voltage across each of the components, $L, C$, and $R$ is $50 \mathrm{~V}$. The voltage across the LC combination will be
(A) $100 \mathrm{~V}$
(B) $50 \sqrt{2} \mathrm{~V}$
(C) $50 \mathrm{~V}$
(D) $0 \mathrm{~V}$ (zero)

Key Concepts

LCR Series Circuit

An LCR series circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in sequence to an AC source. In such circuits, the same current flows through each component, and the total impedance is a combination of the resistive and reactive (inductive and capacitive) impedances. This configuration is fundamental when studying how voltage divides and how phase differences arise between circuit elements.

Phasor Analysis in AC Circuits

Phasor analysis is a method used to represent sinusoidal voltages and currents as rotating vectors (phasors) in the complex plane. This approach converts differential equations into algebraic ones by incorporating both magnitude and phase information, simplifying calculations in AC circuits, especially when dealing with components that introduce phase shifts.

Phase Relationships of Reactive Components

Inductive and capacitive elements in AC circuits cause phase shifts between voltage and current. An inductor creates a voltage that leads the current by 90 degrees, while a capacitor produces a voltage that lags by 90 degrees. This difference in phase is critical when analyzing the net effects in circuits, as opposing phase relationships can result in partial or complete cancellation of voltages.

Resonance and Voltage Cancellation in LC Combinations

When an inductor and a capacitor are combined in series, they can exhibit resonance at a certain frequency where the inductive and capacitive reactances are equal in magnitude but opposite in phase. This resonance causes the reactive voltages to cancel each other out, potentially resulting in a net voltage of zero across the LC combination, despite considerable individual voltages across each component.

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