Four similar prisms of angle of prism are arranged. Which of the following arrangements give no net angular deviation?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining which arrangement of four similar prisms results in no net angular deviation, we can follow these steps:

### Step 1: Understand the Concept of Angular Deviation
The angular deviation (δ) produced by a prism is the angle by which light is deviated from its original path when it passes through the prism. For a single prism, the deviation can be calculated using the formula:
\[ \delta = (n - 1)A \]
where \( n \) is the refractive index of the prism material and \( A \) is the angle of the prism.

### Step 2: Analyze the Arrangements
We need to evaluate the four arrangements of prisms provided in the question. The goal is to find an arrangement where the total deviation (sum of deviations from all prisms) equals zero.

### Step 3: Set Up the Condition for No Net Deviation
For no net angular deviation, the sum of the deviations from the prisms must equal zero:
\[ \delta_1 + \delta_2 + \delta_3 + \delta_4 = 0 \]
This can also be expressed as:
\[ \delta_1 + \delta_2 = -(\delta_3 + \delta_4) \]
This means that the deviations caused by the first two prisms must be equal in magnitude and opposite in direction to the deviations caused by the last two prisms.

### Step 4: Evaluate Each Arrangement
Let’s consider each arrangement (A, B, C, D) and analyze the deviations:
- **Arrangement A**: All prisms arranged in the same direction. This will result in a cumulative deviation and will not yield zero net deviation.
- **Arrangement B**: The first two prisms cause a deviation in one direction, while the last two prisms cause an equal deviation in the opposite direction. This arrangement satisfies the condition for no net deviation.
- **Arrangement C**: Similar to Arrangement A, all prisms in the same direction, leading to a net deviation.
- **Arrangement D**: The arrangement might have prisms that do not cancel out the deviations properly, resulting in a net deviation.

### Step 5: Conclusion
After analyzing the arrangements, we find that **Arrangement B** is the only configuration that results in no net angular deviation.

### Final Answer
The arrangement that gives no net angular deviation is **Arrangement B**.

---