To understand how the resistance of a metal wire varies with its diameter, while keeping other factors like length and temperature constant, we use the formula for resistance:
R== Where:
R is the resistance.
ρ is the resistivity of the material.
ℓ is the length of the wire.
A is the cross-sectional area of the wire.
r is the radius of the wire.
Since
A=πr2, the resistance
R is inversely proportional to the square of the radius:
R∝When considering the diameter
d of the wire, which is twice the radius (
d=2r ), the relation can be rewritten in terms of the diameter:
R∝=This indicates that as the diameter of the wire increases, the resistance decreases quadratically. The relationship between resistance
R and diameter
d is a hyperbolic function, as resistance is inversely proportional to the square of the diameter.