Question

Copper and aluminum are being considered for a high-voltage transmission line that must carry a current of 51.5 A. The resistance per unit length is to be 0.144 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).Jcu= 436.44
`x10^(3) (A)/(m^(2))`

b).λcu=1.05728
`(kg)/(m)`

c).Jal= 274.66
`x10^(3) (A)/(m^(2))`

d).λcu=0.487
`(kg)/(m)`

Step-by-step explanation:

a).

ζcu=1.7
`x10^(-8)`Ωm

ζal=2.76
`x10^(-8)`Ωm

`A=(w)/(R)`

wcu=ζcu*l

`Acu=(1.7x10^(-8)*1000)/(0.144) =1.18x10^(-4) m^(2)`

`J=(I)/(Acu)=(51.5A)/(1.18x10^(-4)m^(2)) \\J=436.44x10^(3) (A)/(m^(2))`

b).

mass per unit Copper

λcu=Dcu*Acu

λcu=8960
`(kg)/(m^(3))`*1.18
`x10^(-4) m^(2)`

λcu=1.05728
`(kg)/(m)`

c).

wal=ζal*l

`Aal=(2.7x10^(-8)*1000)/(0.144) =0.187x10^(-3) m^(2)`

`J=(I)/(Aal)=(51.5A)/(0.187x10^(-3)m^(2)) \\J=274.66x10^(3) (A)/(m^(2))`

d).

mass per unit Aluminum

λal=Dal*Aal

λal=2600
`(kg)/(m^(3))`*0.1875
`x10^(-3) m^(2)`

λcu=0.487
`(kg)/(m)`