In case of four wires of same material, the resistance will be minimum when its length and diameter are respectively :

The resistance of a wire is determined by the formula:

R = ρ * (L / A)

Where:

• ρ is the resistivity of the material (constant for wires made of the same material),
• L is the length of the wire, and
• A is the cross-sectional area of the wire.

The resistance of the wire is directly proportional to its length (L) and inversely proportional to its cross-sectional area (A).

To minimize the resistance of the wire, we need to:

• Minimize the length L, and
• Maximize the cross-sectional area A.

Let’s analyze the given options:

1. Option A: Diameter = D/2 and Length = L/4 
   Cross-sectional area: A = π * (D/2)² = π * D²/4 
   Length: L/4 
   The cross-sectional area is reduced and the length is also reduced, but not to the maximum extent possible.
2. Option B: Diameter = D/4 and Length = 4L 
   Cross-sectional area: A = π * (D/4)² = π * D²/16 
   Length: 4L 
   The cross-sectional area is significantly reduced, and the length is increased, leading to a higher resistance.
3. Option C: Diameter = 2D and Length = L 
   Cross-sectional area: A = π * (2D)² = 4π * D² 
   Length: L 
   The cross-sectional area is maximized, and the length remains the same as given, resulting in the minimum resistance.
4. Option D: Diameter = 4D and Length = 2L 
   Cross-sectional area: A = π * (4D)² = 16π * D² 
   Length: 2L 
   The cross-sectional area is maximized, but the length is doubled, which would increase the resistance.

After analyzing the above options, we can conclude that the resistance will be minimized when the diameter is 2D and the length is L/2. Therefore, the correct answer is:

Correct Option: C