In a prism of small angle A, a beam of light is incident on its surface at an angle of incidence i and leaves the opposite surface of it is angle of `90^(@)` with the surface. If prism material has refractive index of m then the value of angle of incidence will be:

Options

A. A/m
B. A/2m
C. mA
D. mA/2

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry of the Prism
We have a prism with a small angle \( A \). A beam of light is incident on one surface of the prism at an angle of incidence \( i \) and exits the opposite surface at an angle of \( 90^\circ \) with respect to the surface.

### Step 2: Identify the Angles
When the light exits the prism at \( 90^\circ \), it means that the angle of emergence \( E \) is \( 0^\circ \) because it is along the normal.

### Step 3: Use the Angle of Deviation Formula
For a prism, the angle of deviation \( \delta \) can be expressed as:
\[ \delta = i + E - A \]
Since \( E = 0^\circ \), we can simplify this to:
\[ \delta = i - A \]

### Step 4: Relate Angle of Deviation to Refractive Index
For a small angle prism, the angle of deviation can also be expressed in terms of the refractive index \( \mu \):
\[ \delta = (\mu - 1)A \]

### Step 5: Set the Two Expressions for Angle of Deviation Equal
Now we can equate the two expressions for \( \delta \):
\[ i - A = (\mu - 1)A \]

### Step 6: Solve for the Angle of Incidence \( i \)
Rearranging the equation gives:
\[ i = (\mu - 1)A + A \]
\[ i = \mu A \]

### Step 7: Substitute the Refractive Index
Given that the refractive index of the prism material is \( m \), we can substitute \( \mu \) with \( m \):
\[ i = mA \]

### Final Answer
Thus, the value of the angle of incidence \( i \) is:
\[ \boxed{i = mA} \]