A small angled prism of refractive index 1.4 is combined with another small angled prism of refractive index 1.6 to produce disperison without deviation. If the angle of first prism is `6(@)`, then the angle of the second prism is

Options

A. `8^(@)`
B. `6^(@)`
C. `4^(@)`
D. `2^(@)`

Explanation

To solve the problem, we need to find the angle of the second prism that, when combined with the first prism, results in no net deviation while allowing for dispersion. We will use the formula for deviation due to a prism and set up an equation based on the given conditions.

### Step-by-step Solution:

1. **Understanding the Problem:**
We have two prisms:
- Prism 1 with refractive index \( \mu_1 = 1.4 \) and angle \( A_1 = 6^\circ \).
- Prism 2 with refractive index \( \mu_2 = 1.6 \) and angle \( A_2 \) (which we need to find).

The condition is that the total deviation from both prisms should be zero, which means:
\[ (\mu_1 - 1) A_1 + (\mu_2 - 1) A_2 = 0 \]

2. **Substituting Values:**
Substitute the values of \( \mu_1 \), \( A_1 \), and \( \mu_2 \) into the equation:
\[ (1.4 - 1) \cdot 6 + (1.6 - 1) \cdot A_2 = 0 \]
Simplifying this gives:
\[ 0.4 \cdot 6 + 0.6 \cdot A_2 = 0 \]

3. **Calculating the First Term:**
Calculate \( 0.4 \cdot 6 \):
\[ 0.4 \cdot 6 = 2.4 \]
So the equation now looks like:
\[ 2.4 + 0.6 \cdot A_2 = 0 \]

4. **Isolating \( A_2 \):**
Rearranging the equation to solve for \( A_2 \):
\[ 0.6 \cdot A_2 = -2.4 \]
Dividing both sides by \( 0.6 \):
\[ A_2 = \frac{-2.4}{0.6} = -4 \]

5. **Interpreting the Result:**
The negative sign indicates that the second prism must be oriented in the opposite direction to achieve no net deviation. Therefore, the angle of the second prism is:
\[ A_2 = 4^\circ \]

### Final Answer:
The angle of the second prism is \( 4^\circ \).