Q. 164.4( 14 Votes )
Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. (ρAl = 2.63 × 10–8 Ωm, ρCu = 1.72 × 10–8 Ωm, Relative density of Al = 2.7, of Cu = 8.9.)
Resistivity of aluminium, p1 = 2.63 × 10−8 Ω m
Relative density of aluminium, d1 = 2.7
Let l1 be the length of aluminium wire and m1 be its mass.
Resistance of the aluminium wire = R1
Area of cross-section of the aluminium wire = A1
Resistivity of copper, p2 = 1.72 × 10−8 Ω metre
Relative density of copper, d2 = 8.9
Let l2 be the length of copper wire and m2 be its mass.
Resistance of the copper wire = R2
Area of cross-section of the copper wire = A2
We can write,
R1 = (p1 × l1)/A1
R2 = (p2 × l2)/A2
Since R1 = R2
(p1 × l1)/A1 = (p2 × l2)/A2
And l1 = l2
Therefore p1/A1 = p2/A2
A1/A2 = (2.63 × 10-8)/(1.72 × 10–8)
⇒ A1/A2 = 2.63/1.72
Mass of the aluminium wire,
m1 = Volume × Density
= A1 × l1 × d1 = A1 l1 d1 .....1
Mass of the copper wire, m2 = Volume × Density
= A2 × l2 × d2 = A2 l2 d2 .......2
Dividing equation (1) by equation (2), we obtain
m1/m2 = (A1 l1 d1)/(A2 l2 d2)
Since l1 = l2
m1/m2 = A1 d1/A2 d2
Since A1/A2 = 2.63/1.72 calculated above
⇒ m1/m2 = (2.63 × 2.7)/(1.72 × 8.9)
⇒ m1/m2 = 0.46
Since m1 is smaller than m2 Hence, aluminium is lighter than copper.
Since aluminium is lighter, it is preferred for overhead power cables over copper.
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