Question

In an LCR series ac circuit the voltage across each of the components \(\mathrm{L}, \mathrm{C}\) and \(\mathrm{R}\) is \(50 \mathrm{~V}\). The voltage across the LC combination will be (a) \(50 \mathrm{~V}\) (b) \(50 \sqrt{2} \mathrm{~V}\) (c) \(100 \mathrm{~V}\) (d) \(0 \mathrm{~V}\) (zero)

Short answer

The voltage across the LC combination in an LCR series AC circuit with voltages of 50 V across L, C, and R components is 0 V (zero), as the voltages across the inductor (L) and capacitor (C) are out of phase by 180 degrees and cancel each other out when added using phasor addition. The correct answer is (d) \(0 \mathrm{~V}\) (zero).

Step-by-step solution

Identify the Voltages Across Components

The voltage across each of the components is given as 50 V. So, we have: \(V_L = 50 V\) \(V_C = 50 V\) Now, we will find the voltage across the LC combination (V_LC) using phasor addition.

Phasor Addition

In an LCR series AC circuit, the voltages across the inductor (L) and capacitor (C) are out of phase by 180 degrees. To find the resultant voltage across the LC combination, we need to add the voltages of L and C using phasor addition, considering their phase difference. Since they are out of phase by 180 degrees, the total voltage across LC is given by: \(V_{LC} = |V_L - V_C|\)

Calculating Voltage Across LC Combination

Now, we can plug in the values for V_L and V_C and calculate the voltage across the LC combination: \( V_{LC} = |50 V - 50 V|\) \(V_{LC} = |0 V|\) \(V_{LC} = 0 V\) So, the voltage across the LC combination will be 0V (zero). The correct answer is: (d) \(0 \mathrm{~V}\) (zero)

Key concepts

Phasor Addition

Phasor addition is a fascinating concept that simplifies how we handle alternating current (AC) circuit analysis. In AC circuits, voltages and currents are sinusoidal and vary with time, making it challenging to analyze them directly.
Instead, we use phasors, which transform sinusoidal quantities into complex numbers for simpler addition and subtraction. This is particularly useful when dealing with voltages or currents that are out of phase, meaning they don't reach their maximum or minimum values simultaneously.
In the context of an LCR AC circuit, the voltages across the inductor and capacitor are measured using phasors. Since these voltages are 180 degrees out of phase, phasor addition allows us to find the net voltage across the LC combination by essentially "subtracting" the magnitudes, leading to straightforward calculations.

Voltage Across Components

Understanding the voltage across individual components in an LCR circuit is crucial. In our problem, the voltage across each component

• the inductor (L),
• the capacitor (C), and
• the resistor (R)

is given as 50 V. Each component in the circuit responds differently when AC voltage is applied, leading to unique voltages.
The resistor sees a voltage directly proportional to the current, maintaining the same phase. However, the inductor and capacitor voltages are out of phase relative to each current; the inductor voltage leads, while the capacitor voltage lags the current.
Hence, when using phasor addition, the voltages across the inductor and capacitor (considered together as the LC combination) behave differently than those across a resistor.

LC Combination

An LC combination is a fascinating part of an AC circuit. It consists of both an inductor (L) and a capacitor (C). Individually, these components react to changing currents and voltages in opposite ways. An inductor resists changes in current by creating a magnetic field, while a capacitor resists changes in voltage by storing energy as an electric field.
When combined, the voltages across these components can offset each other due to their 180-degree phase difference. In a perfect scenario where their voltages are equal in magnitude, as in our exercise, they cancel each other out completely. This leads to a net voltage across the LC combination of zero, because \[ V_{LC} = |V_L - V_C| = |0| \]is the result of their phase difference.

Phase Difference

Phase difference is key to understanding interactions in AC circuits, especially in an LCR series circuit. It describes the offset in the timing of peaks and troughs between two waveforms, such as voltages or currents.
In our circuit, the voltages across the inductor and capacitor are out of phase by 180 degrees. This phase difference is analogous to one wave reaching its peak while the other reaches its trough.

Understanding this concept helps in predicting how components interact over time. When the inductor voltage leads the circuit by 90 degrees and the capacitor voltage lags by 90 degrees, their combination leads to a complete 180-degree phase difference, allowing us to apply phasor arithmetic efficiently. This interaction is critical to determining how voltages across components balance out or cancel each other, as seen in phasor addition for the LC combination.