Which of the following is a dimensional constant ?

Options

A. Magnification
B. Relative density
C. Gravitational constant
D. Relative error

Answer

To determine which of the given options is a dimensional constant, we need to analyze each option based on whether it has dimensions or not.

### Step-by-Step Solution:

1. **Understanding Dimensional Constants**:
- A dimensional constant is a physical quantity that has dimensions and can be expressed in terms of fundamental units (like mass, length, time, etc.).

2. **Analyzing Each Option**:
- **Option 1: Magnification**
- Magnification is defined as the ratio of the height of the image to the height of the object. Since it is a ratio of two lengths, it is dimensionless.
- **Conclusion**: Not a dimensional constant.

- **Option 2: Relative Density**
- Relative density (or specific gravity) is the ratio of the density of a substance to the density of a reference substance (usually water). Since it is a ratio of two densities, it is also dimensionless.
- **Conclusion**: Not a dimensional constant.

- **Option 3: Gravitational Constant (G)**
- The gravitational constant (G) is a fundamental physical constant used in the calculation of gravitational forces. Its value is approximately \(6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\). This has dimensions of \( [M^{-1} L^3 T^{-2}] \).
- **Conclusion**: This is a dimensional constant.

- **Option 4: Relative Error**
- Relative error is the ratio of the absolute error to the true value. Since it is a ratio of two similar quantities (both having the same units), it is dimensionless.
- **Conclusion**: Not a dimensional constant.

3. **Final Conclusion**:
- Among the given options, the only quantity that has dimensions is the gravitational constant (Option C). Therefore, the correct answer is **Option C: Gravitational Constant**.