Which of the following is a dimensional constant?

Options

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

Answer

To determine which of the given options is a dimensional constant, we need to analyze each option based on its definition and dimensional properties.

### Step-by-step Solution:

1. **Understanding Relative Density**:
- Relative density (also known as specific gravity) is defined as the ratio of the density of a substance to the density of a reference substance (usually water).
- Mathematically, it is expressed as:
\[ \text{Relative Density} = \frac{\text{Density of Substance}}{\text{Density of Reference Substance}} \]
- Since both densities have the same units (mass/volume), they cancel out, making relative density a dimensionless quantity.

**Conclusion**: Relative Density is not a dimensional constant.

2. **Understanding Gravitational Constant (G)**:
- The gravitational constant \( G \) is used in the formula for gravitational force:
\[ F = \frac{G m_1 m_2}{r^2} \]
- Rearranging this gives:
\[ G = \frac{F r^2}{m_1 m_2} \]
- The dimensions of force \( F \) are \( [M L T^{-2}] \), and the dimensions of \( r^2 \) are \( [L^2] \), while \( m_1 \) and \( m_2 \) have dimensions of mass \( [M] \).
- Therefore, the dimensions of \( G \) can be calculated as:
\[ [G] = \frac{[M L T^{-2}] [L^2]}{[M] [M]} = [M^{-1} L^3 T^{-2}] \]
- Since \( G \) has dimensions, it is not a dimensionless constant.

**Conclusion**: Gravitational Constant is not a dimensional constant.

3. **Understanding Refractive Index**:
- The refractive index \( n \) is defined as the ratio of the speed of light in vacuum to the speed of light in a medium:
\[ n = \frac{c_{\text{vacuum}}}{c_{\text{medium}}} \]
- Both speeds are measured in meters per second, so the units cancel out:
\[ n = \frac{\text{m/s}}{\text{m/s}} = 1 \]
- Hence, the refractive index is also dimensionless.

**Conclusion**: Refractive Index is not a dimensional constant.

4. **Understanding Poisson's Ratio**:
- Poisson's ratio \( \nu \) is defined as the ratio of transverse strain to longitudinal strain:
\[ \nu = \frac{\text{Transverse Strain}}{\text{Longitudinal Strain}} = \frac{\Delta d / d}{\Delta l / l} \]
- Since both strains are dimensionless (change in dimension divided by original dimension), Poisson's ratio is also dimensionless.

**Conclusion**: Poisson's Ratio is not a dimensional constant.

### Final Conclusion:
After analyzing all options, the only option that is a dimensional constant is the **Gravitational Constant (G)**.