A spectrometer gives the following reading when used to measure the angle of a prism.
Main scale reading : `58.5 degree`
Vernier scale reading : `09` divisions
Given that `1` division on main scale correspods to `0.5` degree. Total divisions on the vernier scale is `30` and match with `29` divisions of the main scale. the angle of the prism from the above data:

Options

A. `58.59^(@)`
B. `58.77^(@)`
C. `58.65^(@)`
D. `59^(@)`

Answer

To find the angle of the prism using the readings from the spectrometer, we can follow these steps:

### Step 1: Write down the given data
- Main Scale Reading (MSR) = 58.5 degrees
- Vernier Scale Reading (VSR) = 09 divisions
- 1 division on the Main Scale (MS) = 0.5 degrees
- Total divisions on the Vernier Scale (VS) = 30
- Vernier Scale matches with 29 divisions of the Main Scale.

### Step 2: Calculate the value of 1 Vernier Scale Division (VSD)
The relationship between the Vernier Scale and Main Scale can be expressed as:
\[ 1 \text{ VSD} = \frac{29 \text{ MSD}}{30} \]
Substituting the value of 1 MSD:
\[ 1 \text{ VSD} = \frac{29}{30} \times 0.5 \text{ degrees} = \frac{14.5}{30} \text{ degrees} = 0.4833 \text{ degrees} \]

### Step 3: Calculate the Least Count (LC)
The Least Count is given by the formula:
\[ \text{Least Count} = 1 \text{ MSD} - 1 \text{ VSD} \]
Substituting the values:
\[ \text{Least Count} = 0.5 \text{ degrees} - 0.4833 \text{ degrees} = 0.0167 \text{ degrees} \]

### Step 4: Calculate the final reading
The final reading can be calculated using the formula:
\[ \text{Final Reading} = \text{MSR} + (\text{VSR} \times \text{LC}) \]
Substituting the values:
\[ \text{Final Reading} = 58.5 \text{ degrees} + (9 \times 0.0167 \text{ degrees}) = 58.5 \text{ degrees} + 0.1503 \text{ degrees} = 58.6503 \text{ degrees} \]

### Step 5: Round off the final reading
The angle of the prism is approximately:
\[ \text{Angle of the Prism} \approx 58.65 \text{ degrees} \]

### Final Answer
The angle of the prism is **58.65 degrees**.

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