Most materials have the refractive index, `n gt 1`. So, when a light ray from air enters a naturally occuring material, then by Snell's law, `(sin theta_(1))/(sin theta_(2)) = (n_(1))/(n_(2))`, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, `n = (c // v) = +- sqrt(epsilon_(r ), mu_(r ))`, where `c` is the speed of the electromagnetic waves in vacuum, `v` its speed in the medium, `epsilon_( r)` and `mu_( r)` are negative, one must choose the negative root of `n`. Such negative refractive index materials can now be artifically prepared and are called meta-materials. They exhibit significantly different optical behaviour, without violating any physical laws. Since `n` is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Answer the following questions :
Choose the correct statement.

Options

A. The speed of ligth in the meta-material is `v = c |n|`
B. The speed of ligth in the meta-material is `v =( c)/ (|n|)`
C. The speed of ligth in the meta-material is `v = c`
D. The wavelength of the light in the meta-material `(lambda_(m))` is given by `lambda_(m) = lambda_(air) |n|`, Where `lambda_(air) is the wavelength of the ligth in air

Explanation

To solve the question regarding the behavior of light in metamaterials and the correct statements about the speed of light and wavelength, we can break down the solution into the following steps:

### Step 1: Understand the Refractive Index
The refractive index \( n \) of a material is defined as the ratio of the speed of light in vacuum \( c \) to the speed of light in the material \( v \):
\[ n = \frac{c}{v} \]
For most materials, \( n > 1 \), meaning that light slows down when entering the material. However, in metamaterials, the refractive index can be negative.

### Step 2: Analyze the Speed of Light in Metamaterials
For metamaterials with a negative refractive index, we can express the speed of light in the material as:
\[ v = \frac{c}{n} \]
Since \( n \) is negative, we can take the absolute value to ensure speed remains positive:
\[ v = \frac{c}{|n|} \]
This indicates that the speed of light in a metamaterial is inversely proportional to the absolute value of the refractive index.

### Step 3: Evaluate the Given Statements
1. **Speed of light in metamaterial is \( v = c \times |n| \)**: This is incorrect because it suggests that speed increases with increasing refractive index, which contradicts the definition.
2. **Speed of light in metamaterial is \( v = \frac{c}{|n|} \)**: This is correct as derived above.
3. **Speed of light in metamaterial is \( v = c \)**: This is incorrect since the speed of light changes in a medium.
4. **Wavelength in metamaterial is \( \lambda = \lambda_0 \times n \)**: This is incorrect because the wavelength in a medium is given by:
\[ \lambda = \frac{\lambda_0}{|n|} \]
where \( \lambda_0 \) is the wavelength in vacuum.

### Step 4: Conclusion
From the analysis, the correct statement is:
- **Speed of light in metamaterial is \( v = \frac{c}{|n|} \)**.