Answer:
a) The magnitude JJ of the current density for a copper cable is 5.91 × 10⁵A.m⁻²
b)The mass per unit length \lambdaλ for a copper cable is 0.757kg/m
c)The magnitude J of the current density for an aluminum cable is 3.5 × 10⁵A/m²
d)The mass per unit length \lambdaλ for an aluminum cable is 0.380kg/m
Explanation:
The expression for electric field of conductor is,
![E = \frac{V}{L}](https://tex.z-dn.net/?f=E%20%3D%20%20%5Cfrac%7BV%7D%7BL%7D)
The general equation of voltage is,
V = iR
The expression for current density in term of electric field is,
![J = \frac{E}{p}](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7BE%7D%7Bp%7D)
Substitute (V/L) for E in the above equation of current density.
![J = \frac{V}{pL} ------(1)](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7BV%7D%7BpL%7D%20------%281%29)
Substitute iR for V in equation (1)
![J = \frac{iR}{pL} ------(2)](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7BiR%7D%7BpL%7D%20------%282%29)
Substitute 1.69 × 10⁸ Ω .m for p
50A for i
0.200Ω.km⁻¹ for (R/L) in eqn (2)
![J = \frac{(50) (0.200\times 10^-^3) }{1.69 \times 10^-^8 } \\\\= 5.91 \times 10^5A.m^-^2](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7B%2850%29%20%280.200%5Ctimes%2010%5E-%5E3%29%20%7D%7B1.69%20%5Ctimes%2010%5E-%5E8%20%7D%20%5C%5C%5C%5C%3D%205.91%20%5Ctimes%2010%5E5A.m%5E-%5E2)
The magnitude JJ of the current density for a copper cable is 5.91 × 10⁵A.m⁻²
b) The expression for resistivity of the conductor is,
![p = \frac{RA}{L}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7BRA%7D%7BL%7D)
![A = \frac{pL}{R}](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7BpL%7D%7BR%7D)
The expression for mass density of copper is,
m = dV
where, V is the density of the copper.
Substitute AL for V in equation of the mass density of copper.
m=d(AL)
m/L = dA
λ is use for (m/L)
substitute,
pL/R for A and λ is use for (m/L) in the eqn above
![\lambda = d\frac{p}{\frac{R}{L} } ------(3)](https://tex.z-dn.net/?f=%5Clambda%20%3D%20d%5Cfrac%7Bp%7D%7B%5Cfrac%7BR%7D%7BL%7D%20%7D%20------%283%29)
Substitute 0.200Ω.km⁻¹ for (R/L)
8960kgm⁻³ for d and 1.69 × 10⁸ Ω .m
![\lambda = (8960) \frac{(1.69 \times 10^-^8 }{0.200\times 10^-^3} \\\\= 0.757kg.m^-^1](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%288960%29%20%5Cfrac%7B%281.69%20%5Ctimes%2010%5E-%5E8%20%7D%7B0.200%5Ctimes%2010%5E-%5E3%7D%20%5C%5C%5C%5C%3D%200.757kg.m%5E-%5E1)
c) Using the equation (2) current density for aluminum cable is,
![J = \frac{iR}{pL}](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7BiR%7D%7BpL%7D)
p is the resistivity of the aluminum cable.
Substitute 2.82 × 10⁻⁸Ω.m for p ,
50A for i and 0.200Ω.km⁻¹ for (R/L)
![J = \frac{(50)(0.200\times10^-^3) }{2.89\times 10^-^8} \\\\= 3.5 \times10^5A/m^2](https://tex.z-dn.net/?f=J%20%3D%20%5Cfrac%7B%2850%29%280.200%5Ctimes10%5E-%5E3%29%20%7D%7B2.89%5Ctimes%2010%5E-%5E8%7D%20%5C%5C%5C%5C%3D%203.5%20%5Ctimes10%5E5A%2Fm%5E2)
The magnitude J of the current density for an aluminum cable is 3.5 × 10⁵A/m²
d) Using the equation (3) mass per unit length for aluminum cable is,
![\lambda = d\frac{p}{\frac{R}{L} }](https://tex.z-dn.net/?f=%5Clambda%20%3D%20d%5Cfrac%7Bp%7D%7B%5Cfrac%7BR%7D%7BL%7D%20%7D)
p is the resistivity and is the density of the aluminum cable.
Substitute 0.200Ω.km⁻¹ for (R/L), 2700 for d and 2.82 × 10⁻⁸Ω.m for p
![\lambda = (2700) \frac{(2.82 \times 10^-^8) }{(0.200 \times 10^-^3) } \\\\= 0.380kg/m](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%282700%29%20%5Cfrac%7B%282.82%20%5Ctimes%2010%5E-%5E8%29%20%7D%7B%280.200%20%5Ctimes%2010%5E-%5E3%29%20%7D%20%5C%5C%5C%5C%3D%200.380kg%2Fm)
The mass per unit length \lambdaλ for an aluminum cable is 0.380kg/m