If `R,X_L, and X_C` represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

Options

A. `R X_L X_C`
B. `R/(sqrt(X_L X_C))`
C. `R/(sqrt(X_L/ X_C))`
D. `vR X_L/X_C`

Answer

To solve the problem, we need to analyze the dimensions of the given quantities: resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \).

### Step-by-Step Solution:

1. **Identify the Dimensions of Each Quantity**:
- The resistance \( R \) has the dimension of ohms (Ω).
- The inductive reactance \( X_L \) also has the dimension of ohms (Ω).
- The capacitive reactance \( X_C \) also has the dimension of ohms (Ω).

Thus, we can summarize:
\[ [R] = [X_L] = [X_C] = \Omega \]

2. **Understanding the Relationship**:
- Since \( R \), \( X_L \), and \( X_C \) all have the same dimensions (ohms), we can use them in ratios to determine if any combination is dimensionless.

3. **Check the Expression**:
- We are asked to check the expression \( \frac{R}{\sqrt{X_L \cdot X_C}} \).
- First, calculate the dimensions of \( X_L \cdot X_C \):
\[ [X_L \cdot X_C] = [X_L] \cdot [X_C] = \Omega \cdot \Omega = \Omega^2 \]
- Now, take the square root:
\[ [\sqrt{X_L \cdot X_C}] = \sqrt{\Omega^2} = \Omega \]

4. **Calculate the Final Expression**:
- Now substitute back into the expression:
\[ \left[\frac{R}{\sqrt{X_L \cdot X_C}}\right] = \frac{[R]}{[\sqrt{X_L \cdot X_C}]} = \frac{\Omega}{\Omega} = 1 \]
- Since the result is 1, it is dimensionless.

5. **Conclusion**:
- The expression \( \frac{R}{\sqrt{X_L \cdot X_C}} \) is dimensionless.

### Final Answer:
The dimensionless quantity is \( \frac{R}{\sqrt{X_L \cdot X_C}} \).