The dimensional formula for refractive index is

Options

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

Answer

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

### Step 1: Understand the definition of refractive index
The refractive index (n) of a medium is defined as the ratio of the speed of light in vacuum (c) to the speed of light in that medium (v). Mathematically, it can be expressed as:
\[ n = \frac{c}{v} \]

### Step 2: Identify the units of speed
The speed of light in vacuum (c) and the speed of light in a medium (v) both have the same units, which are meters per second (m/s).

### Step 3: Write down the dimensions of speed
The dimensional formula for speed (which is distance/time) is given by:
\[ [\text{Speed}] = [L T^{-1}] \]
where \( L \) represents length and \( T \) represents time.

### Step 4: Substitute the dimensions into the refractive index formula
Since both speeds (c and v) have the same dimensional formula, we can substitute this into the refractive index formula:
\[ n = \frac{[c]}{[v]} = \frac{[L T^{-1}]}{[L T^{-1}]} \]

### Step 5: Simplify the expression
When we divide the dimensions, we have:
\[ n = \frac{[L T^{-1}]}{[L T^{-1}]} = [L^0 T^0] = M^0 L^0 T^0 \]
This indicates that the refractive index is a dimensionless quantity.

### Conclusion
Thus, the dimensional formula for the refractive index is:
\[ M^0 L^0 T^0 \]

### Final Answer
The correct option is \( M^0 L^0 T^0 \).

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