A crown glass prism of refracting angle `8^(@)` is combined with a flint glass prism to obtain deviation without dispersion. If the refractive indicates for red and violet rays for the crown glass are 1.514 and 1.524 and for the flint glass are 1.645 and 1.665 respectivey, find the angle of flint glass prism and net deviation.

Options

A. `3^@`
B. `4^@`
C. `4.5^@`
D. `5^@`

Answer

To solve the problem step by step, we will break it down into parts: finding the angle of the flint glass prism and then calculating the net deviation.

### Step 1: Understanding the Problem
We need to combine a crown glass prism with a flint glass prism such that the total deviation of light is achieved without any dispersion. This means that the dispersion caused by the crown glass prism must be equal to the dispersion caused by the flint glass prism.

### Step 2: Define Dispersion
Dispersion is defined as the difference in refractive indices for violet (µv) and red (µr) light multiplied by the angle of the prism (A). The formula for dispersion is:
\[ \text{Dispersion} = (\mu_v - \mu_r) \cdot A \]

### Step 3: Calculate Dispersion for Crown Glass Prism
Given:
- Refractive index for crown glass:
- µv (violet) = 1.524
- µr (red) = 1.514
- Angle of crown glass prism (A) = 8°

Using the dispersion formula for crown glass:
\[ D_c = (1.524 - 1.514) \cdot 8 \]
\[ D_c = 0.01 \cdot 8 = 0.08 \]

### Step 4: Set Up the Equation for Flint Glass Prism
Let the angle of the flint glass prism be A'. The dispersion for the flint glass prism is given by:
- Refractive index for flint glass:
- µv (violet) = 1.665
- µr (red) = 1.645

Using the dispersion formula for flint glass:
\[ D_f = (1.665 - 1.645) \cdot A' \]
\[ D_f = 0.02 \cdot A' \]

### Step 5: Equate the Dispersion
Since we want the total dispersion to be zero:
\[ D_c = D_f \]
\[ 0.08 = 0.02 \cdot A' \]

### Step 6: Solve for A'
Now, we can solve for A':
\[ A' = \frac{0.08}{0.02} = 4° \]

### Step 7: Calculate the Net Deviation
To find the net deviation (D), we use the formula:
\[ D = |(\mu - 1) \cdot A - (\mu' - 1) \cdot A'| \]
Where:
- µ is the average refractive index for crown glass
- µ' is the average refractive index for flint glass

### Step 8: Calculate Average Refractive Indices
For crown glass:
\[ \mu = \frac{1.514 + 1.524}{2} = 1.519 \]

For flint glass:
\[ \mu' = \frac{1.645 + 1.665}{2} = 1.655 \]

### Step 9: Substitute Values into the Deviation Formula
Now substituting the values into the deviation formula:
\[ D = |(1.519 - 1) \cdot 8 - (1.655 - 1) \cdot 4| \]
\[ D = |0.519 \cdot 8 - 0.655 \cdot 4| \]
\[ D = |4.152 - 2.62| \]
\[ D = |1.532| \]

### Final Answers
1. The angle of the flint glass prism (A') is **4°**.
2. The net deviation (D) is approximately **1.53°**.