To find the diameter of the wire using the screw gauge, we need to understand the relationship between the main scale and the circular scale.$\\$Given:$\\$• Main scale reading (MSR) $= 0$ mm$\\$• Circular scale reading (CSR) $= 52$ divisions$\\$• 1 mm on the main scale corresponds to 100 divisions on the circular scale.$\\$The formula to calculate the diameter using a screw gauge is:$\\$Diameter$=$MSR$+ (\frac{CSR \times Least Count}{1}) \\$The least count (LC) of the screw gauge is the value of one division on the circular scale, which is:$\\$LC$= \frac{1 mm}{100 divisions} = 0.01$ mm$\\$Substitute the values into the formula:$\\$Diameter$= 0$ mm$+ (\frac{52 \times 0.01 mm}{1}) \\$Calculate:$\\$Diameter$= 0 + 0.52$ mm$\\$Convert the diameter from mm to cm:$\\$Diameter$= 0.52$ mm$= \frac{0.52}{10}$ cm$= 0.052$ cm$\\$Therefore, the diameter of the wire is $0.052$ cm, which corresponds to Option 4.