Differences in Resistance of Copper Wires with Different Diameters

The relationship between the diameter of copper wires and their electrical resistance is a critical topic in electrical engineering and physics. This article explores how the diameter of two copper wires affects their resistance when the length is kept constant. The principle involved is rooted in the resistivity formula, a fundamental concept in electrical engineering that helps us understand and predict electrical behavior.

Basic Principles of Electrical Resistance

Electrical resistance is a measure of how an object opposes the flow of electric current. This opposition is directly related to the material's resistivity, the length of the object, and the cross-sectional area. The formula for resistance is given by:

R ρ(L / A)

Where:

R is the resistance in Ohms ρ (rho) is the resistivity of the material in Ohm-meters (Ω·m) L is the length of the conductor in meters (m) A is the cross-sectional area of the conductor in square meters (m2)

The resistivity (ρ) of copper is a constant value, which means the only variable in the formula is the area (A). The area is directly proportional to the square of the diameter (D). Therefore, the area (A) can be expressed as:

A πr2 π(D/2)2 πD2/4

Comparing Two Copper Wires with Different Diameters

Lets consider two copper wires of the same length, with diameters in the ratio 1:2. Let the diameter of the first wire be D1 2r1 and the diameter of the second wire be D2 2r2, where r1 r and r2 2r.

The area of the first wire (A1) will be:

A1 π(D1/2)2 π(2r)2/4 πr2

The area of the second wire (A2) will be:

A2 π(D2/2)2 π(4r)2/4 4πr2

Now, the ratio of their areas (A2/A1) is:

A2/A1 (4πr2) / (πr2) 4

Since the resistance is inversely proportional to the cross-sectional area, the ratio of their resistances (R1/R2) will be the inverse of the ratio of their areas:

R2/R1 A1/A2 1/4

Therefore, the resistance of the second wire is four times that of the first wire.

Conclusion

The example above provides a clear illustration of how the diameter of copper wires directly influences their resistance. When the diameter of a copper wire doubles, the resistance is reduced to one-fourth. This principle is essential in electrical engineering, particularly in the design and analysis of electrical circuits.