Which of the following pairs have the same dimensions ?
(L = inductance , C = capacitance , R = resistance)

Options

A. `(L)/( R ) and CR`
B. ` LR and CR`
C. `(L)/( R) and sqrt(L C)`
D. `R C and (1)/( L C )`

Explanation

To determine which pairs have the same dimensions among the given options (L = inductance, C = capacitance, R = resistance), we will analyze each pair step by step.

### Step-by-Step Solution:

1. **Understanding the Dimensions**:
- Inductance (L) has the dimension: \([L^2 T^{-2} A^{-2}]\)
- Capacitance (C) has the dimension: \([L^{-2} T^{4} A^{2}]\)
- Resistance (R) has the dimension: \([L^{2} T^{-3} A^{-2}]\)

2. **Analyzing the First Pair: \( \frac{L}{R} \)**
- The dimension of \( \frac{L}{R} \):
\[ \frac{[L^2 T^{-2} A^{-2}]}{[L^{2} T^{-3} A^{-2}]} = [T] \]
- This has the dimension of time.

3. **Analyzing the Second Pair: \( CR \)**
- The dimension of \( CR \):
\[ [L^{-2} T^{4} A^{2}] \cdot [L^{2} T^{-3} A^{-2}] = [T] \]
- This also has the dimension of time.

4. **Analyzing the Third Pair: \( LR \)**
- The dimension of \( LR \):
\[ [L^2 T^{-2} A^{-2}] \cdot [L^{2} T^{-3} A^{-2}] = [L^{4} T^{-5} A^{-4}] \]
- This does not have the dimension of time.

5. **Analyzing the Fourth Pair: \( \frac{L}{R} \) and \( \sqrt{LC} \)**
- We already calculated \( \frac{L}{R} \) has the dimension of time.
- Now calculating \( \sqrt{LC} \):
\[ \sqrt{[L^2 T^{-2} A^{-2}] \cdot [L^{-2} T^{4} A^{2}]} = \sqrt{[T^{2}]} = [T] \]
- This has the dimension of time.

6. **Analyzing the Fifth Pair: \( RC \)**
- The dimension of \( RC \):
\[ [L^{2} T^{-3} A^{-2}] \cdot [L^{-2} T^{4} A^{2}] = [T] \]
- This also has the dimension of time.

7. **Analyzing the Last Pair: \( \frac{1}{LC} \)**
- The dimension of \( \frac{1}{LC} \):
\[ \frac{1}{[L^2 T^{-2} A^{-2}] \cdot [L^{-2} T^{4} A^{2}]} = \frac{1}{[T^{2}]} = [T^{-2}] \]
- This does not have the dimension of time.

### Conclusion:
The pairs that have the same dimensions (time) are:
- \( \frac{L}{R} \) and \( CR \)
- \( \frac{L}{R} \) and \( \sqrt{LC} \)
- \( CR \) and \( RC \)

### Final Answer:
The pairs that have the same dimensions are:
1. \( \frac{L}{R} \) and \( CR \)
2. \( \frac{L}{R} \) and \( \sqrt{LC} \)
3. \( CR \) and \( RC \)