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 Refractive Index**:
The refractive index (n) is defined as the ratio of the speed of light in a medium (v) to the speed of light in a vacuum (c):
\[ n = \frac{v}{c} \]

2. **Identifying the Dimensions of Speed**:
The speed of light in any medium or in a vacuum has the same dimensional formula. Speed is defined as distance traveled per unit time. Therefore, the dimensional formula for speed (v or c) is:
\[ [v] = [c] = LT^{-1} \]
where L represents length and T represents time.

3. **Calculating the Dimensional Formula for Refractive Index**:
Since both the numerator and denominator in the refractive index formula have the same dimensions, we can express the dimensional formula of the refractive index as:
\[ [n] = \frac{[v]}{[c]} = \frac{LT^{-1}}{LT^{-1}} = 1 \]
This means that the refractive index is dimensionless.

4. **Considering Relative Refractive Index**:
The relative refractive index (often denoted as \( n_{21} \)) is defined as the ratio of the refractive index of two different media:
\[ n_{21} = \frac{n_2}{n_1} \]
Since both \( n_2 \) and \( n_1 \) are dimensionless (as established), the relative refractive index is also dimensionless:
\[ [n_{21}] = \frac{[n_2]}{[n_1]} = \frac{1}{1} = 1 \]

5. **Conclusion**:
Therefore, the dimensional formula for the relative refractive index is:
\[ [n_{21}] = M^0 L^0 T^0 \]
which indicates that it is a dimensionless quantity.

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

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