In an LCR series ac circuit , the voltage across each of the components , L,C and R is 50 V . The voltage across the LC combination will be :

Options

A. 50 V
B. `50sqrt(2)` V
C. 100 V
D. 0 V (zero)

Answer

To solve the problem of finding the voltage across the LC combination in an LCR series AC circuit where the voltage across each component (L, C, and R) is 50 V, we can follow these steps:

### Step 1: Understand the Circuit
In a series LCR circuit, the voltage across the resistor (R), inductor (L), and capacitor (C) can be represented as:
- Voltage across R: \( V_R = 50 \, V \)
- Voltage across L: \( V_L = 50 \, V \)
- Voltage across C: \( V_C = 50 \, V \)

### Step 2: Use Phasor Representation
In an AC circuit, the voltages can be represented as phasors. The voltage across the resistor is in phase with the current, while the voltages across the inductor and capacitor are out of phase with the current:
- \( V_R \) is in phase with the current.
- \( V_L \) leads the current by \( 90^\circ \) (or \( \frac{\pi}{2} \) radians).
- \( V_C \) lags the current by \( 90^\circ \) (or \( \frac{\pi}{2} \) radians).

### Step 3: Write the Voltage Equations
Using the phasor representation:
- \( V_R = 50 \, V \)
- \( V_L = 50 \, V \, \text{(leading)} \)
- \( V_C = 50 \, V \, \text{(lagging)} \)

### Step 4: Calculate the Voltage Across LC Combination
The total voltage across the LC combination can be calculated using the fact that the voltages across L and C are equal in magnitude but opposite in phase:
- \( V_{LC} = V_L - V_C \)

Since both \( V_L \) and \( V_C \) are 50 V, we can express this as:
- \( V_{LC} = 50 \, V \, (leading) - 50 \, V \, (lagging) \)

### Step 5: Simplify the Expression
Since \( V_L \) leads \( V_C \) by \( 180^\circ \) (or \( \pi \) radians), we can write:
- \( V_{LC} = 50 \, V - 50 \, V = 0 \, V \)

### Conclusion
The voltage across the LC combination is:
\[ V_{LC} = 0 \, V \]