A ray of light is incident on the surface of a glass plate of index of refraction 1.55 at the polarizing angle. Calculate the angle of refraction.

Answer

To solve the problem of finding the angle of refraction when a ray of light is incident on a glass plate at the polarizing angle, we can follow these steps:

### Step 1: Understand the Polarizing Angle
The polarizing angle (also known as Brewster's angle) is given by the formula:
\[ \tan(i) = n \]
where \( n \) is the refractive index of the medium (in this case, glass).

### Step 2: Calculate the Polarizing Angle
Given that the refractive index \( n \) of the glass is 1.55, we can calculate the polarizing angle \( i \):
\[ i = \tan^{-1}(n) = \tan^{-1}(1.55) \]

Using a calculator:
\[ i \approx 57.17^\circ \]

### Step 3: Apply Snell's Law
According to Snell's Law:
\[ n_1 \sin(i) = n_2 \sin(r) \]
In this case, we can assume that the incident medium is air, where \( n_1 = 1 \) and \( n_2 = 1.55 \).

### Step 4: Use the Relationship Between Angles
At the polarizing angle, the relationship between the angle of incidence \( i \) and the angle of refraction \( r \) is:
\[ i + r = 90^\circ \]
From this, we can express the angle of refraction \( r \) as:
\[ r = 90^\circ - i \]

### Step 5: Calculate the Angle of Refraction
Substituting the value of \( i \):
\[ r = 90^\circ - 57.17^\circ \]
\[ r \approx 32.83^\circ \]

### Final Answer
The angle of refraction \( r \) is approximately \( 32.83^\circ \).

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