The dimensional formula for relative refractive index is

Options

A. `[M^(1)L^(1)T^(1)]`
B. `[M^(0)L^(0)T^(0)]`
C. `[M^(1)L^(0)T^(0)]`
D. `[MLT^(-1)]`

Answer

To find the dimensional formula for the relative refractive index, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relative Refractive Index**:
The relative refractive index (n) is defined as the ratio of the speed of light in one medium to the speed of light in another medium. It can be expressed mathematically as:
\[ n = \frac{c_1}{c_2} \]
where \(c_1\) is the speed of light in the first medium and \(c_2\) is the speed of light in the second medium.

2. **Dimensional Formula of Velocity**:
The dimensional formula for velocity (speed) is given by:
\[ [V] = [L][T]^{-1} \]
where [L] represents length and [T] represents time.

3. **Applying the Dimensional Formula**:
Since the relative refractive index is a ratio of two velocities, we can write:
\[ n = \frac{c_1}{c_2} \Rightarrow [n] = \frac{[c_1]}{[c_2]} = \frac{[L][T]^{-1}}{[L][T]^{-1}} \]

4. **Cancelling the Dimensions**:
When we divide the two dimensional formulas, we find:
\[ [n] = \frac{[L][T]^{-1}}{[L][T]^{-1}} = 1 \]
This means that the dimensions cancel out completely.

5. **Conclusion**:
Since the relative refractive index is a ratio of two quantities with the same dimensions, it is dimensionless. Therefore, the dimensional formula for the relative refractive index is:
\[ [n] = M^0 L^0 T^0 \]

### Final Answer:
The dimensional formula for relative refractive index is \(M^0 L^0 T^0\) (dimensionless).

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