The correct option is It becomes one-fourth.
Explanation
The electrical resistance (R) of a cylindrical conductor is governed by its physical dimensions and the intrinsic property of the material known as resistivity. Resistance is directly proportional to the length of the conductor and inversely proportional to its cross-sectional area.
Mathematical Analysis:
The formula for resistance is given by:
R = ρ (L / A)
- ρ (rho): Resistivity of the material
- L: Length of the wire
- A: Cross-sectional area
For a wire with diameter d, the cross-sectional area A is calculated as:
A = π (d / 2)² = (π d²) / 4
Substituting this into the resistance formula shows the relationship between resistance and diameter:
R ∝ 1 / d² (Resistance is inversely proportional to the square of the diameter)
If the diameter is doubled (becomes 2d):
- The new area becomes proportional to (2d)² = 4d².
- Consequently, the area increases by a factor of 4.
- Since resistance is inversely proportional to the area, the new resistance becomes one-fourth of the original value.
Key Takeaway:
Doubling the diameter of a wire quadruples its cross-sectional area, which results in the resistance decreasing to one-fourth of its initial value, provided the length and material remain unchanged.