The dimensional formula for refractive index is,

A. [ M^0 L^1 T^-2 ]

B. [ M^0 L^0 T^0 ]

C. [ M^0 L^1 T^1 ]

D. [ M^1 L^1 T^-1 ]

The refractive index is a fundamental concept in optics that describes how light propagates through different media. To determine the dimensional formula for the refractive index, we first need to understand what it represents. The refractive index (n) is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in a medium (v). Mathematically, this is expressed as:

n = c / v

Breaking Down the Components

To analyze the dimensional formula, we need to consider the dimensions of speed. Speed is defined as distance traveled per unit time. Therefore, the dimensional formula for speed (v) can be expressed as:

Speed (v) = Distance / Time

In terms of dimensions, this translates to:

• Distance has the dimension of length, denoted as [L].
• Time has the dimension of time, denoted as [T].

Thus, the dimensional formula for speed is:

[L T^-1]

Applying to the Refractive Index

Now, since the refractive index is the ratio of two speeds (the speed of light in a vacuum and the speed of light in a medium), we can express it as:

n = [L T^-1] / [L T^-1]

When we divide these two quantities, the length dimensions cancel out, and the time dimensions also cancel out:

n = [L T^-1] / [L T^-1] = [M^0 L^0 T^0]

Conclusion on Dimensional Formula

As a result, the dimensional formula for the refractive index is:

[M^0 L^0 T^0]

This indicates that the refractive index is a dimensionless quantity, meaning it has no units. Therefore, the correct answer to your question is:

B. [M^0 L^0 T^0]

Why It Matters

Understanding that the refractive index is dimensionless is crucial in optics, as it allows for comparisons between different materials without the complications of units. This concept is widely applied in various fields, including physics, engineering, and even in everyday applications like lenses and optical fibers.