Which of the following dimensions will be the same as that of time?

Options

A. `L/R`
B. `C/L`
C. LC
D. `R/L`

Answer

To solve the problem of determining which dimension is the same as that of time, we will analyze each option provided in the question step by step.

### Step-by-Step Solution:

1. **Identify the Dimension of Time:**
- The dimension of time is represented as \( [T] \).

2. **Analyze Option 1: \( \frac{L}{R} \) (Inductance/Resistance):**
- The dimension of inductance \( L \) is given as \( [M L^2 T^{-2} I^{-2}] \).
- The dimension of resistance \( R \) is given as \( [M L^2 T^{-3} I^{-2}] \).
- Now, calculate the dimension of \( \frac{L}{R} \):
\[ \frac{[M L^2 T^{-2} I^{-2}]}{[M L^2 T^{-3} I^{-2}]} = [T] \]
- The resultant dimension is \( [T] \), which is the same as the dimension of time.

3. **Analyze Option 2: \( \frac{C}{L} \) (Capacitance/Inductance):**
- The dimension of capacitance \( C \) is given as \( [M^{-1} L^{-2} T^4 I^2] \).
- The dimension of inductance \( L \) is \( [M L^2 T^{-2} I^{-2}] \).
- Now, calculate the dimension of \( \frac{C}{L} \):
\[ \frac{[M^{-1} L^{-2} T^4 I^2]}{[M L^2 T^{-2} I^{-2}]} = [M^{-2} L^{-4} T^6 I^4] \]
- The resultant dimension is not the same as time.

4. **Analyze Option 3: \( LC \) (Inductance × Capacitance):**
- Using the dimensions from previous steps:
\[ [L] = [M L^2 T^{-2} I^{-2}] \]
\[ [C] = [M^{-1} L^{-2} T^4 I^2] \]
- Now, calculate the dimension of \( LC \):
\[ [L] \times [C] = [M L^2 T^{-2} I^{-2}] \times [M^{-1} L^{-2} T^4 I^2] = [T^2] \]
- The resultant dimension is \( [T^2] \), which is not the same as time.

5. **Analyze Option 4: \( \frac{R}{L} \) (Resistance/Inductance):**
- Using the dimensions from previous steps:
\[ [R] = [M L^2 T^{-3} I^{-2}] \]
\[ [L] = [M L^2 T^{-2} I^{-2}] \]
- Now, calculate the dimension of \( \frac{R}{L} \):
\[ \frac{[M L^2 T^{-3} I^{-2}]}{[M L^2 T^{-2} I^{-2}]} = [T^{-1}] \]
- The resultant dimension is \( [T^{-1}] \), which is not the same as time.

### Conclusion:
- The only option that has the same dimension as time is **Option 1: \( \frac{L}{R} \)**.