Question

A copper cable needs to carry a current of 160 A with a power loss of only 2.0 W/m. What is the required radius of the copper cable? (The resistivity of copper is 1.7 × 10 −8 Ω⋅m).

Answer 1

The radius of the cable is 0.0083 m or 8.3 mm.

Step-by-step explanation:

The resistance of copper cable of 1 meter length will be given by

`R_(cable) = (\rho * l)/(a)` .... (i)

where the resistivity of copper is
`\rho = 1.7 * 10^(-8) \Omega.m` , and l is the length of the wire which is considered to be 1m, and a is the cross sectional area of the wire in
`m^(2)`.

From the formula of power we know that,
`P = I^2 R` .... (ii)

Therefore 2 W/m =
`(160)^2 * R` .... (iii)

where the resistance,R, actually means the resistance of the cable per meter.

Therefore R ( resistance of cable per meter)

=
`(2)/(160^2) = 7.812 * 10^(-5)` ohms / meter. .... (iv)

Therefore from (i)

`7.812 * 10^(-5)` =
`(1.7 * 10^(-8) * 1)/(a) = (1.7 * 10^(-8) * 1)/(\pi r^(2) )` ..... (v)

where cross sectional area of the cable, a =
`\pi r^2`,

where r is the radius of the cable, and length of cable,l = 1m.

Therefore r =
`\sqrt{( 1.7 * 10^(-8))/(\pi * 7.812 * 10^(-5) ) } = 0.0083m = 8.3 mm`