Question

One wire possesses a solid core of copper, with a circular cross-section of radius 3.78 mm. The other wire is composed of 19 strands of thin copper wire bundled together. Each strand has a circular cross-section of radius 0.756 mm. The current density J in each wire is the same, J=2950 A/m².
1. How much current does each wire carry?
2. The resistivity of copper is 1.69 x 10⁻⁸ ohm m. What is the resistance of a 1.00 m length of each wire?

Answer 1

Answer:

a) Current in wire 1 = 0.132 A

Current in wire 2 = 0.101 A

b) Resistance of wire 1 = R₁ = 0.000376 Ω = (3.76 × 10⁻⁴) Ω = 0.376 mΩ

Resistance of wire 2 = R₂ = 0.000495 Ω = (4.95 × 10⁻⁴) Ω = 0.495 mΩ

Explanation:

Current density, J = (current) × (cross sectional area)

Current density for both wires = J = 2950 A/m²

For wire 1,

Cross sectional Area = πr² = π(0.00378²)

A₁ = 0.00004491 m²

For wire 2,

With the assumption that the strands are well banded together with no spaces in btw.

Cross sectional Area = 19 × πr² = π(0.000756)²

A₂ = 0.00003413 m²

Current in wire 1 = I₁ = J × A₁ = 2950 × 0.00004491 = 0.132 A

Current in wire 2 = I₂ = J × A₂ = 2950 × 0.00003413 = 0.101 A

b) Resistance = ρL/A

ρ = resistivity for both wires = (1.69 x 10⁻⁸) Ω.m

L = length of wire = 1.00 m for each of the two wires

A₁ = 0.00004491 m²

A₂ = 0.00003413 m²

R₁ = ρL/A₁ = (1.69 x 10⁻⁸ × 1)/0.00004491

R₁ = 0.000376 Ω = (3.76 × 10⁻⁴) Ω = 0.376 mΩ

R₂ = ρL/A₂ = (1.69 x 10⁻⁸ × 1)/0.00003413

R₂ = 0.000495 Ω = (4.95 × 10⁻⁴) Ω = 0.495 mΩ

Hope this helps!!