R = (rho)*l/A where rho is the resistivity, l is the length of wire and A is the cross sectional area
A = pi*r^2 = pi*d^2/4

R = (rho)*l/(pi*d^2/4)
If d is doubled
d' = 2d
R' = (rho)*l/(pi*(2d)^2/4)
R' = (rho)*l/(pi*d^2)
R'/R = [(rho)*l/(pi*d^2) ]/ [ (rho)*l/(pi*d^2/4)]
R'R = 1/4
So, the resistance of the wire has decreased to 1/4th of its original one

**How will the resistance of a wire change if its diameter is doubled, its length remaining the same?**

**Introduction:**
Resistance is a property of a wire that determines the opposition it offers to the flow of electric current. It depends on various factors, including the length, cross-sectional area, and resistivity of the wire material. In this case, we are examining the effect of doubling the diameter (and therefore the radius) of a wire while keeping its length constant on its resistance.

**Explanation:**
To understand how the resistance changes when the diameter of a wire is doubled, we need to consider the relationship between resistance, diameter, and other relevant parameters.

**1. Relationship between resistance and diameter:**
The resistance of a wire is inversely proportional to the cross-sectional area of the wire. When the diameter of a wire is doubled, its cross-sectional area increases by a factor of four (since area is proportional to the square of the radius). Consequently, the resistance decreases by a factor of four.

**2. Relationship between resistance and length:**
The resistance of a wire is directly proportional to its length. If the length of the wire remains the same, the resistance will also remain unchanged.

**3. Applying the changes:**
In this scenario, the diameter of the wire is doubled, resulting in a fourfold increase in its cross-sectional area. However, the length of the wire remains the same. As a result, the resistance of the wire will decrease by a factor of four due to the increase in the cross-sectional area.

**Conclusion:**
In summary, when the diameter of a wire is doubled while its length remains constant, the resistance of the wire decreases by a factor of four. This is because the resistance is inversely proportional to the cross-sectional area of the wire. It is important to note that this relationship holds true as long as the wire material and temperature remain constant.