Question

Two insulated copper wires of similar overall diameter have very different interiors. One wire possesses a solid core of copper, with a circular cross section of radius 2.28 mm. The other wire is composed of 19 strands of thin copper wire bundled together. Each strand has a circular cross section of radius 0.456 mm. The current density J in each wire is the same. J = 3750 A /m^2
a. How much current does each wire carry?
b. What is the resistance of a 6.25 m length of each wire?

Answer 1

Answer with Explanation:

We are given that

Radius of  solid core wire=r=2.28 mm=$2.28\times 10^{-3} m$

$1mm=10^{-3} m$

Radius of each strand  of thin wire=r'=0.456 mm=$0.456\times 10^{-3} m$

Current density of each wire=$J=3750 A/m^2$

a.Area =$\pi r^2$

Where $\pi=3.14$

Using the formula

Cross section area of copper wire has solid core =$3.14\times (2.28\times 10^{-3})^2=16.3\times 10^{-6} m^2$

Current density =$J=\frac{I}{A}$

Using the formula

$3750=\frac{I}{16.3\times 10^{-6}}$

$I=3750\times 16.3\times 10^{-6}=0.061 A$

Total number of strands=19

Area of strand wire=$A'=19\times 3.14\times (0.456\times 10^{-3})^2=12.4\times 10^{-6} m^2$

$J'=\frac{I'}{A'}$

$3750=\frac{I'}{19\times 3.14(0.456\times 10^{-3})^2}$

$I'=3750\times 19\times 3.14(0.456\times 10^{-3})^2$

$I'=0.047 A$

b.Resistivity of copper wire=$\rho=1.69\times 10^{-8}\Omega-m$

Length of each wire =6.25 m

Resistance, R=$\frac{\rho l}{A}$

Using the formula

Resistance of solid core wire=$R=\frac{1.69\times 10^{-8}\times 6.25}{16.3\times 10^{-6}}=6.5\times 10^{-3}\Omega$

Resistance of strand wire=$R'=\frac{1.69\times 10^{-8}\times 6.25}{12.4\times 10^{-6}}=8.5\times 10^{-3}\Omega$