The pitch of a screw gauge is `1mm` and its cap is divided into `100` divisions. When this screw gauge is used to measure the diameter of a wire, then main scale reading is `2 mm` and `58^(th)` division of circular scale coincides with the main scale. There is no zero error in the screw gage. Then the diameter of the wire is

Options

A. `2.38 mm`
B. `2.48 mm`
C. `2.58 mm`
D. `3.58 mm`

Explanation

To find the diameter of the wire using the screw gauge, we can follow these steps:

### Step 1: Identify the given data
- **Pitch of the screw gauge (P)** = 1 mm
- **Number of divisions on the circular scale (N)** = 100
- **Main scale reading (MSR)** = 2 mm
- **Circular scale reading (CSR)** = 58th division
- **Zero error** = 0 (no zero error)

### Step 2: Calculate the least count (LC) of the screw gauge
The least count is calculated using the formula:
\[ \text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{P}{N} \]
Substituting the values:
\[ LC = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} \]

### Step 3: Calculate the total reading
The total reading (diameter of the wire) can be calculated using the formula:
\[ \text{Diameter} = \text{Main Scale Reading (MSR)} + (\text{Least Count (LC)} \times \text{Circular Scale Reading (CSR)}) \]
Substituting the values:
\[ \text{Diameter} = 2 \text{ mm} + (0.01 \text{ mm} \times 58) \]
Calculating the multiplication:
\[ 0.01 \text{ mm} \times 58 = 0.58 \text{ mm} \]
Now, adding this to the main scale reading:
\[ \text{Diameter} = 2 \text{ mm} + 0.58 \text{ mm} = 2.58 \text{ mm} \]

### Final Answer
The diameter of the wire is **2.58 mm**.

---