Question

An aluminum power transmission line has a resistance of 0.0580 Ω / km .

a) What is its mass per kilometer?

(b) What is the mass per kilometer of a copper line having the same resistance?

c)A lower resistance would shorten the heating time. Discuss the practical limits to speeding the heating by lowering the resistance.

Answer 1

Here, we need to use the relation between resistance and resistivity

R = $\rho$L / A ------------- (1)

where $\rho$ is the resistivity of the material

so,

mass per unit length = m / L

we know mass = density (d) * volume (V)

where volume (V) = area of cross section (A) * length (L)

so,

Let's denote mass per unit length as X.

X = m/L

X = dAL / L ------------ (2)

L = RA / $\rho$ -------------- (from (1))

Put this in (2)

so

(a) X = dL$\rho$ / R

we are given

R/L as 0.0580 ohm/ km = 5.8e-5 ohm / m

we need L / R, so 1 / 5.8e-5

Now, the density of aluminium is 2600 Kg / m³ and resistivity of aluminium is 2.75e-8 ohm /m

so,

X = 2600 * 1 / 5.8e-5 * 2.75e-8

X = 1.2327 Kg / m

we need mass per kilometer.

so,

X = 1.23e3 Kg / Km ( Aluminium)

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(b) For copper, the density is 8960 Kg / m³ and resistivity is 1.69e-8 ohm /m

X = 8960 * 1 / 5.8e-5 * 1.69e-8

X = 2.6e3 kg / Km  (copper)

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(c) yes, if we decrease the resistance, more current will flow which will heat up the wires in lesser time. There is a limit , however. The resistivity of material will determine how much resistance can be decreased so that material of wire does not get damaged due to excessive heating.