The dispersive powers of flint glass and crown glass are 0.053 and 0.034 respectively and their mean refractive indices are 1.68 and 1.53 for white light .Calculate the angle of the flint glass prism required to form an achromatic combination with a crown glass prism of refracting anlge `4^@` .

Options

A. `2^@`
B. `4^@`
C. `5^@`
D. `6^@`

Answer

To solve the problem of finding the angle of the flint glass prism required to form an achromatic combination with a crown glass prism of refracting angle \(4^\circ\), we will follow these steps:

### Step 1: Understand the Given Data
We have the following data:
- Dispersive power of flint glass (\(\omega_f\)) = 0.053
- Dispersive power of crown glass (\(\omega_c\)) = 0.034
- Mean refractive index of flint glass (\(n_f\)) = 1.68
- Mean refractive index of crown glass (\(n_c\)) = 1.53
- Angle of crown glass prism (\(A_c\)) = \(4^\circ\)

### Step 2: Use the Achromatic Condition
For an achromatic combination, the condition is given by:
\[ \omega_f (n_w - 1) A_f = \omega_c (n_w' - 1) A_c \]
Where:
- \(n_w\) is the refractive index for the blue light.
- \(n_w'\) is the refractive index for the red light.
- \(A_f\) is the angle of the flint glass prism.

### Step 3: Express the Dispersion in Terms of Refractive Indices
The dispersive power is defined as:
\[ \omega = \frac{n_b - n_r}{n - 1} \]
Where:
- \(n_b\) is the refractive index for blue light.
- \(n_r\) is the refractive index for red light.
- \(n\) is the mean refractive index.

From this, we can express \(n_b - n_r\) as:
\[ n_b - n_r = \omega (n - 1) \]

### Step 4: Substitute the Values
Using the values for flint and crown glass:
- For flint glass:
\[ n_b - n_r = \omega_f (n_f - 1) = 0.053 \times (1.68 - 1) = 0.053 \times 0.68 = 0.03604 \]
- For crown glass:
\[ n_b' - n_r' = \omega_c (n_c - 1) = 0.034 \times (1.53 - 1) = 0.034 \times 0.53 = 0.01802 \]

### Step 5: Set Up the Equation
Now substituting into the achromatic condition:
\[ 0.053 \times 0.68 \times A_f = 0.034 \times 0.53 \times 4 \]
This simplifies to:
\[ 0.03604 \times A_f = 0.01802 \times 4 \]

### Step 6: Solve for \(A_f\)
Calculating the right side:
\[ 0.01802 \times 4 = 0.07208 \]
Now we can solve for \(A_f\):
\[ A_f = \frac{0.07208}{0.03604} = 2 \]

### Final Answer
The angle of the flint glass prism required to form an achromatic combination with the crown glass prism is:
\[ \boxed{2^\circ} \]