The index of refraction can be defined as the velocity of light In a vacuum divided by the velocity In the medium `(N=C/S)` If this is the case, another valid expression for Snell's law Is

Options

A. `(Sin theta_i)/(Sin theta_r)=C_r/S_r`
B. `(Sin theta_i)/(Sin theta_r)=C_i/C_r`
C. `S_i/S_r=N_r/N_i`
D. `(Sin theta_i)/(Sin theta_r)=S_i/S_r`

Explanation

To derive another valid expression for Snell's law using the index of refraction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Index of Refraction**:
The index of refraction (N) is defined as:
\[ N = \frac{C}{S} \]
where \(C\) is the speed of light in a vacuum and \(S\) is the speed of light in the medium.

2. **Consider Two Media**:
Let’s consider two media with indices of refraction \(N_1\) and \(N_2\). The speed of light in these media can be expressed as:
\[ S_1 = \frac{C}{N_1} \quad \text{and} \quad S_2 = \frac{C}{N_2} \]

3. **Using Snell's Law**:
Snell's law states that:
\[ N_1 \sin(\theta_i) = N_2 \sin(\theta_r) \]
where \(\theta_i\) is the angle of incidence and \(\theta_r\) is the angle of refraction.

4. **Substituting the Speeds**:
We can express \(N_1\) and \(N_2\) in terms of the speeds of light:
\[ N_1 = \frac{C}{S_1} \quad \text{and} \quad N_2 = \frac{C}{S_2} \]
Substituting these into Snell's law gives:
\[ \frac{C}{S_1} \sin(\theta_i) = \frac{C}{S_2} \sin(\theta_r) \]

5. **Canceling \(C\)**:
Since \(C\) is a common factor on both sides, we can cancel it out:
\[ \frac{\sin(\theta_i)}{S_1} = \frac{\sin(\theta_r)}{S_2} \]

6. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{\sin(\theta_i)}{\sin(\theta_r)} = \frac{S_1}{S_2} \]

7. **Final Expression**:
This can be interpreted as:
\[ \frac{\sin(\theta_i)}{\sin(\theta_r)} = \frac{N_2}{N_1} \]
which is another valid expression for Snell's law.