Given:
Diameter of the copper wire = 0.5 mm
Resistivity of copper = 1.6 × 10^-8 ohm meter
Desired resistance = 10 ohm

To find:
Length of the wire to achieve a resistance of 10 ohm
Change in resistance if the diameter is doubled

Solution:

1. Finding the length of the wire:
Resistance can be calculated using the formula:
Resistance = (Resistivity * Length) / Area

The area of the wire can be calculated using the formula:
Area = π * (Diameter/2)^2

Given that the diameter is 0.5 mm, we can calculate the area as follows:
Area = π * (0.5/2)^2 = π * (0.25)^2 = 0.1963 mm^2

Converting the area from mm^2 to m^2:
Area = 0.1963 × 10^-6 m^2

Now, we can rearrange the resistance formula to find the length:
Length = (Resistance * Area) / Resistivity
Length = (10 * 0.1963 × 10^-6) / (1.6 × 10^-8)
Length = 1.225 m

Therefore, the length of the wire required to achieve a resistance of 10 ohm is approximately 1.225 meters.

2. Change in resistance if the diameter is doubled:
Resistance is inversely proportional to the area of the wire. If the diameter is doubled, the area of the wire will increase by a factor of 4.

Using the area formula mentioned above, we can find the new area for a doubled diameter:
New Area = π * ((2 * Diameter)/2)^2
New Area = π * (Diameter)^2 * 4

We can see that the new area is 4 times the original area. Therefore, the resistance will decrease by a factor of 4.

So, if the diameter is doubled, the resistance will decrease to 1/4th of its original value.

Summary:
- The length of the wire required to achieve a resistance of 10 ohm is approximately 1.225 meters.
- If the diameter of the wire is doubled, the resistance will decrease to 1/4th of its original value.