A thin prism having refreacting angle `10^(@)` is made of galss refractive index 1.42. This prism is combined with another thin prism glass of refractive index 1.7 This Combination profuces dispersion without deviation. The refreacting angle of second prishm should be

Options

A. `4^(@)`
B. `6^(@)`
C. `8^(@)`
D. `10^(@)`

Answer

To solve the problem, we need to find the refracting angle of the second prism (A2) such that the combination of the two prisms produces dispersion without deviation. Here’s the step-by-step solution:

### Step 1: Understand the Condition of Dispersion without Deviation
For the combination of two prisms to produce dispersion without deviation, the total deviation (Δ) must be zero. This means that the deviation produced by the first prism (Δ1) must be equal in magnitude but opposite in sign to the deviation produced by the second prism (Δ2).

### Step 2: Write the Formula for Deviation
The formula for the deviation (Δ) produced by a prism is given by:
\[ \Delta = A \cdot (n - 1) \]
where:
- A is the refracting angle of the prism,
- n is the refractive index of the prism.

### Step 3: Set Up the Equation
For the first prism:
- Refracting angle (A1) = 10 degrees
- Refractive index (n1) = 1.42

The deviation produced by the first prism (Δ1) can be calculated as:
\[ \Delta_1 = A_1 \cdot (n_1 - 1) = 10 \cdot (1.42 - 1) \]

For the second prism:
- Refractive index (n2) = 1.7
- Refracting angle (A2) = ?

The deviation produced by the second prism (Δ2) is:
\[ \Delta_2 = A_2 \cdot (n_2 - 1) = A_2 \cdot (1.7 - 1) \]

### Step 4: Set the Total Deviation to Zero
Since the total deviation is zero, we have:
\[ \Delta_1 + \Delta_2 = 0 \]
This implies:
\[ \Delta_1 = -\Delta_2 \]

Substituting the expressions for Δ1 and Δ2:
\[ 10 \cdot (1.42 - 1) = -A_2 \cdot (1.7 - 1) \]

### Step 5: Calculate Δ1
Calculating Δ1:
\[ \Delta_1 = 10 \cdot 0.42 = 4.2 \]

### Step 6: Substitute Δ1 into the Equation
Now substituting Δ1 into the equation:
\[ 4.2 = -A_2 \cdot 0.7 \]

### Step 7: Solve for A2
Rearranging the equation to solve for A2:
\[ A_2 = -\frac{4.2}{0.7} = -6 \]

### Final Answer
The refracting angle of the second prism (A2) should be:
\[ A_2 = -6 \text{ degrees} \]