Which of the following is a dimensional constant?

Options

A. Refractive index
B. Poisson's ratio
C. Relative density
D. Gravitational constant

Explanation

To determine which of the given options is a dimensional constant, we will analyze each option step by step.

### Step 1: Analyze the Refractive Index
- The refractive index (μ) is defined as the ratio of the speed of light in vacuum to the speed of light in a medium.
- Mathematically, it is expressed as:
\[ \mu = \frac{c}{v} \]
where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the medium.
- Since it is a ratio of two speeds, it is dimensionless (it has no units).
- **Conclusion**: The refractive index is NOT a dimensional constant.

### Step 2: Analyze Poisson's Ratio
- Poisson's ratio (σ) is defined as the ratio of lateral strain to longitudinal strain.
- Mathematically, it is expressed as:
\[ \sigma = \frac{\text{lateral strain}}{\text{longitudinal strain}} \]
- Both lateral strain and longitudinal strain are dimensionless quantities (they are ratios of lengths).
- **Conclusion**: Poisson's ratio is NOT a dimensional constant.

### Step 3: Analyze Relative Density
- Relative density (specific gravity) is defined as the ratio of the density of a substance to the density of water.
- Mathematically, it is expressed as:
\[ \text{Relative Density} = \frac{\text{Density of substance}}{\text{Density of water}} \]
- Since it is a ratio of two densities, it is also dimensionless.
- **Conclusion**: Relative density is NOT a dimensional constant.

### Step 4: Analyze Gravitational Constant
- The gravitational constant (G) is used in the formula for gravitational force:
\[ F = \frac{G m_1 m_2}{r^2} \]
- Rearranging gives:
\[ G = \frac{F r^2}{m_1 m_2} \]
- The dimensions of force (F) are \( [M L T^{-2}] \), and the dimensions of distance squared (\(r^2\)) are \( [L^2] \). The dimensions of mass (\(m\)) are \( [M] \).
- Thus, the dimensions of G can be calculated as:
\[ [G] = \frac{[M L T^{-2}] [L^2]}{[M]^2} = [M^{-1} L^3 T^{-2}] \]
- **Conclusion**: The gravitational constant has dimensions and is considered a dimensional constant.

### Final Conclusion
Among the options analyzed, the only dimensional constant is the **gravitational constant (G)**.

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