A thin prism P with angle `4^(@)` and made from glass of refractive index 1.54 is combined with another thin prism P made from glass of refractive index 1.72 to produce dispersion without deviation The angle of prism P is

Options

A. `2.6^(@)`
B. `3^(@)`
C. `4^(@)`
D. `5.33^(@)`

Answer

To solve the problem, we need to determine the angle of the second prism \( A_2 \) that, when combined with the first prism \( A_1 \), results in no net deviation while producing dispersion.

### Step-by-Step Solution:

1. **Identify Given Values:**
- For the first prism \( P_1 \):
- Angle \( A_1 = 4^\circ \)
- Refractive index \( \mu_1 = 1.54 \)
- For the second prism \( P_2 \):
- Refractive index \( \mu_2 = 1.72 \)
- Angle \( A_2 \) is what we need to find.

2. **Understand the Condition for No Deviation:**
- The condition for no net deviation when combining two prisms is given by:
\[ (\mu_1 - 1) A_1 + (\mu_2 - 1) A_2 = 0 \]
- This means that the deviation caused by the first prism must be equal and opposite to the deviation caused by the second prism.

3. **Substitute the Known Values:**
- Substitute the known values into the equation:
\[ (1.54 - 1) \cdot 4 + (1.72 - 1) \cdot A_2 = 0 \]
- Simplifying this gives:
\[ 0.54 \cdot 4 + 0.72 \cdot A_2 = 0 \]

4. **Calculate the First Prism's Contribution:**
- Calculate \( 0.54 \cdot 4 \):
\[ 0.54 \cdot 4 = 2.16 \]
- The equation now becomes:
\[ 2.16 + 0.72 \cdot A_2 = 0 \]

5. **Solve for \( A_2 \):**
- Rearranging the equation gives:
\[ 0.72 \cdot A_2 = -2.16 \]
- Dividing both sides by \( 0.72 \):
\[ A_2 = \frac{-2.16}{0.72} \]
- Calculating this gives:
\[ A_2 = -3^\circ \]
- Since the angle cannot be negative in this context, we take the absolute value:
\[ A_2 = 3^\circ \]

### Final Answer:
The angle of the second prism \( A_2 \) is \( 3^\circ \).