Question

Copper and aluminum are being considered for a high-voltage transmission line that must carry a current of 56.4 A. The resistance per unit length is to be 0.181 Ω/km. The densities of copper and aluminum are 8960 and 2600 kg/m3, respectively. Compute (a) the magnitude J of the current density and (b) the mass per unit length λ for a copper cable and (c) J and (d) λ for an aluminum cable.

Answer 1

a)
`J=6.04* 10^5\ A/m^2`

b)λ=0.83 kg/m

c)
`J=3.71* 10^5\ A/m^2`

d) λ=0.39 kg/m

Step-by-step explanation:

Given that

I= 56.4 A

R= 0.181 Ω/km

Copper :

d=8960 kg/m³

`\rho = 1.69* 10^(-8)\ ohm.m`

Aluminium :

d=2600 kg/m³

`\rho = 2.75* 10^(-8)\ ohm.m`

We know that

E= V/L

V= I R

J= E/ρ

`J=(IR)/(L\rho)`

For copper:

`J=(IR)/(L\rho)`

`J=(56.4* 0.181* 10^(-3))/(1.69* 10^(-8))`

`J=6.04* 10^5\ A/m^2`

Mass per unit length λ

`\lambda =(\rho d)/((R)/(L))`

`\lambda =(1.69* 10^(-8)* 8960 )/(0.181* 10^(-3))`

λ=0.83 kg/m

For aluminum :

`J=(IR)/(L\rho)`

`J=(56.4* 0.181* 10^(-3))/(2.75* 10^(-8))`

`J=3.71* 10^5\ A/m^2`

Mass per unit length λ

`\lambda =(\rho d)/((R)/(L))`

`\lambda =( 2.75* 10^(-8)* 2600  )/(0.181* 10^(-3))`

λ=0.39 kg/m