In a series LCR circuit, the the rms voltage supply is 170 V. If `V_(R), V_(L) and V_(C)` represent the rms voltage drop across resistor , inductor and capacitor, then

Options

A. `V_(R) le 170V` (always)
B. `V_(L) ge 170 V` (always)
C. `V_(L) or V_(C)` may be greater than 170 V
D. `|V_(L)-V_(C)| lt 170`

Answer

To solve the problem regarding the series LCR circuit with an RMS voltage supply of 170 V, we will analyze the relationships between the RMS voltage drops across the resistor (V_R), inductor (V_L), and capacitor (V_C).

### Step-by-Step Solution:

1. **Understanding the Phasor Diagram**:
In a series LCR circuit, the voltage across the resistor (V_R) is in phase with the current, while the voltages across the inductor (V_L) and capacitor (V_C) are out of phase with the current. The phasor diagram shows V_R along the horizontal axis, V_L upwards, and V_C downwards.

2. **Applying Kirchhoff's Voltage Law**:
According to Kirchhoff's law, the total voltage in the circuit is given by:
\[ V = V_R + (V_L - V_C) \]
However, since V_L and V_C are out of phase, we need to consider their vector nature.

3. **Using the Pythagorean Theorem**:
The effective voltage (V) in the circuit can be expressed as:
\[ V^2 = V_R^2 + (V_L - V_C)^2 \]
Given that the RMS voltage supply is 170 V, we have:
\[ 170^2 = V_R^2 + (V_L - V_C)^2 \]

4. **Deriving Relationships**:
- **For V_R**: Since \(V_R^2\) and \((V_L - V_C)^2\) are both positive, it follows that:
\[ V_R^2 \leq 170^2 \implies V_R \leq 170 \text{ V} \]
- **For V_L and V_C**: The difference \(V_L - V_C\) can also be analyzed. If we set \(V_R = 0\), we find:
\[ 170^2 = (V_L - V_C)^2 \implies |V_L - V_C| = 170 \text{ V} \]
This means that either \(V_L\) or \(V_C\) can be greater than 170 V, depending on their values.

5. **Conclusion**:
- \(V_R \leq 170\) V (always true)
- \(V_L\) or \(V_C\) can be greater than 170 V (possible)
- The difference \(V_L - V_C\) can vary, but it can be less than or equal to 170 V depending on the values of \(V_L\) and \(V_C\).

### Summary of Relationships:
1. \(V_R \leq 170\)
2. \(V_L\) or \(V_C\) may be greater than 170.
3. \(V_L - V_C\) can be less than or equal to 170.