If `L,C` and `R` denote the inductance, capacitance and resistance respectively, the dimensional formula for `C^(2)LR` is

Options

A. `[ML^(-2)T^(-1)I^(0)]`
B. `[M^(0)L^(0)T^(3)I^(0)]`
C. `[M^(-1)L^(-2)T^(6)I^(2)]`
D. `[M^(0)L^(0)T^(2)I^(0)]`

Answer

To find the dimensional formula for \( C^2LR \), where \( L \) is inductance, \( C \) is capacitance, and \( R \) is resistance, we will derive the dimensions of each component and then combine them accordingly.

### Step-by-Step Solution:

1. **Identify the dimensions of each component:**
- The dimensional formula for **inductance (L)** is given by:
\[ [L] = [M^1 L^2 T^{-2} I^{-2}] \]
- The dimensional formula for **capacitance (C)** is given by:
\[ [C] = [M^{-1} L^{-2} T^4 I^2] \]
- The dimensional formula for **resistance (R)** is given by:
\[ [R] = [M^1 L^2 T^{-3} I^{-2}] \]

2. **Calculate \( C^2 \):**
- To find the dimensions of \( C^2 \):
\[ [C^2] = ([C])^2 = [M^{-1} L^{-2} T^4 I^2]^2 = [M^{-2} L^{-4} T^8 I^4] \]

3. **Combine the dimensions of \( C^2 \), \( L \), and \( R \):**
- Now we will combine the dimensions:
\[ [C^2LR] = [C^2] \cdot [L] \cdot [R] \]
- Substitute the dimensional formulas:
\[ [C^2LR] = [M^{-2} L^{-4} T^8 I^4] \cdot [M^1 L^2 T^{-2} I^{-2}] \cdot [M^1 L^2 T^{-3} I^{-2}] \]

4. **Combine the dimensions step by step:**
- First, combine \( [C^2] \) and \( [L] \):
\[ [C^2L] = [M^{-2} L^{-4} T^8 I^4] \cdot [M^1 L^2 T^{-2} I^{-2}] = [M^{-1} L^{-2} T^6 I^2] \]
- Now combine this result with \( [R] \):
\[ [C^2LR] = [M^{-1} L^{-2} T^6 I^2] \cdot [M^1 L^2 T^{-3} I^{-2}] = [M^{0} L^{0} T^{3} I^{0}] = [T^3] \]

5. **Final Result:**
- The dimensional formula for \( C^2LR \) is:
\[ [C^2LR] = [T^3] \]

### Conclusion:
The dimensional formula for \( C^2LR \) is \( T^3 \).