Introduction:
When a piece of aluminum wire of finite length is drawn through a series of dies to reduce its diameter to half its original value, its resistance will change. This change in resistance can be explained by considering the relationship between resistance and the dimensions of a wire.

Resistance and Dimensions:
Resistance is directly proportional to the length of a wire and inversely proportional to its cross-sectional area. The resistance of a wire can be calculated using the formula R = ρ * (L/A), where R is resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.

Effect of Diameter Reduction:
When the diameter of the aluminum wire is reduced to half its original value, its cross-sectional area decreases by a factor of four. This is because the cross-sectional area of a wire is proportional to the square of its diameter. Therefore, if the diameter is halved, the area becomes one-fourth of its original value.

Resistance Calculation:
Using the resistance formula, we can calculate the new resistance of the aluminum wire after diameter reduction. Let's assume the original resistance of the wire is R1, original length is L1, original diameter is D1, and the resistivity of aluminum is ρ.

1. Calculate the original cross-sectional area:
- Original radius (R1) = D1/2
- Original area (A1) = π * R1^2

2. Calculate the new cross-sectional area:
- New radius (R2) = D1/4 (half of the original diameter)
- New area (A2) = π * R2^2

3. Calculate the new length:
- New length (L2) = L1 (since the length remains unchanged)

4. Calculate the new resistance:
- New resistance (R2) = ρ * (L2/A2)

Conclusion:
In conclusion, when a piece of aluminum wire of finite length is drawn through a series of dies to reduce its diameter to half its original value, its resistance will increase. This is due to the decrease in the cross-sectional area of the wire, which leads to a higher resistance according to the resistance formula.