A ray of light travelling from first medium is incident at an angle of `30^(@)` on the second medium and refracted at an angle of `45^(@)`. Find refractive index of the second medium with respect to first medium and angle of emergence.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a ray of light traveling from the first medium to the second medium. The angle of incidence in the first medium is \(30^\circ\), and the angle of refraction in the second medium is \(45^\circ\). We need to find the refractive index of the second medium with respect to the first medium and the angle of emergence.

### Step 2: Use Snell's Law
Snell's Law states that:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
Where:
- \(n_1\) is the refractive index of the first medium,
- \(\theta_1\) is the angle of incidence,
- \(n_2\) is the refractive index of the second medium,
- \(\theta_2\) is the angle of refraction.

Assuming the refractive index of the first medium (\(n_1\)) is 1 (for air), we can write:
\[ \sin(30^\circ) = n_2 \sin(45^\circ) \]

### Step 3: Calculate the Sine Values
Using known values:
- \(\sin(30^\circ) = \frac{1}{2}\)
- \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\)

Substituting these values into the equation gives:
\[ \frac{1}{2} = n_2 \cdot \frac{1}{\sqrt{2}} \]

### Step 4: Solve for \(n_2\)
Rearranging the equation to solve for \(n_2\):
\[ n_2 = \frac{1/2}{1/\sqrt{2}} = \frac{1}{2} \cdot \sqrt{2} = \frac{\sqrt{2}}{2} \]

### Step 5: Find the Angle of Emergence
When the ray exits from the second medium back into the first, we again apply Snell's Law:
\[ n_2 \sin(\theta_2) = n_1 \sin(E) \]
Where \(E\) is the angle of emergence.

Substituting the known values:
\[ \frac{\sqrt{2}}{2} \sin(45^\circ) = 1 \cdot \sin(E) \]

Using \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\):
\[ \frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{2}} = \sin(E) \]
\[ \frac{1}{2} = \sin(E) \]

### Step 6: Calculate \(E\)
To find \(E\), we take the inverse sine:
\[ E = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \]

### Final Results
- The refractive index of the second medium with respect to the first medium is \(\frac{\sqrt{2}}{2}\).
- The angle of emergence \(E\) is \(30^\circ\).

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