Choose the false statement

Options

A. Relative permittivity is a dimensionless constant
B. Angular displacement has neither units not dimensions
C. Refractive index is dimensionless variable
D. Permeability of vaccum is a dimensional constant

Answer

To solve the question of identifying the false statement among the given options, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Understanding Relative Permittivity**:
- Relative permittivity (also known as dielectric constant) is defined as the ratio of the permittivity of a substance to the permittivity of free space (vacuum).
- Since it is a ratio of two similar quantities, it is dimensionless.
- **Conclusion**: This statement is true.

2. **Analyzing Angular Displacement**:
- Angular displacement is defined as the angle through which an object has rotated about a specific axis.
- It is measured in radians, which is a dimensionless unit because it represents a ratio of two lengths (arc length to radius).
- However, it does have a unit (radians) but is often considered dimensionless in terms of physical dimensions.
- **Conclusion**: This statement needs careful consideration, but it is generally accepted that angular displacement has units (radians) and is dimensionless in terms of physical dimensions.

3. **Examining Refractive Index**:
- The refractive index is defined as the ratio of the speed of light in vacuum to the speed of light in a medium.
- As it is a ratio of two speeds, it is also dimensionless.
- **Conclusion**: This statement is true.

4. **Considering Permeability of Vacuum**:
- The permeability of vacuum (also known as magnetic permeability) is a measure of how much resistance is encountered when forming a magnetic field in a vacuum.
- It is a constant with a specific value (approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \)), which means it has dimensions (specifically, it has units).
- **Conclusion**: This statement is false.

### Final Answer:
The false statement among the options provided is: **"Angular displacement has neither units nor dimensions."**