To determine the diameter of the wire using the screw gauge, we need to consider both the pitch scale reading and the circular scale reading. The formula to calculate the diameter is:

$ D = P + (C \times L) $

Where:

• $D$ is the diameter of the wire.
• $P$ is the pitch scale reading.
• $C$ is the circular scale reading.
• $L$ is the least count of the screw gauge.

Given:

• Pitch of the screw gauge, $P = 1 \mathrm{~mm}$
• Number of divisions on the circular scale = 100
• Pitch scale reading, $P = 1 \mathrm{~mm}$
• The circular scale reads $63^{\text{rd}}$ division.

First, we calculate the least count of the screw gauge:

$$ L = \frac{ \text{Pitch} }{ \text{Number of divisions} } = \frac{ 1 \mathrm{~mm} }{ 100 } = 0.01 \mathrm{~mm} $$

Next, we substitute the values into the formula for the diameter:

$$ D = 1 \mathrm{~mm} + (63 \times 0.01 \mathrm{~mm}) $$

Calculate the product:

$$ 63 \times 0.01 \mathrm{~mm} = 0.63 \mathrm{~mm} $$

Now, add the pitch scale reading and the circular scale reading:

$$ D = 1 \mathrm{~mm} + 0.63 \mathrm{~mm} = 1.63 \mathrm{~mm} $$

Convert $1.63 \mathrm{~mm}$ to centimeters:

$$ 1.63 \mathrm{~mm} = 0.163 \mathrm{~cm} $$

Therefore, the diameter of the wire is:

Option B: $0.163 \mathrm{~cm}$