Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? - Physics

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Numerical

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.
Al = 2.63 × 10−8 Ω m, ρCu = 1.72 × 10−8 Ω m, Relative density of Al = 2.7, of Cu = 8.9.)

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Solution

Resistivity of aluminium, ρAl = 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, ρCu = 1.72 × 10−8 Ω m

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

The two relations can be written as

`"R"_1 = ρ_1"l"_1/"A"_1` ..........(1)

`"R"_2 = ρ_2"l"_2/"A"_2` ............(2)

It is given that 

R1 = R2

∴ `ρ_1"l"_1/"A"_1 = ρ_2"l"_2/"A"_2`

And 

I1 = I2

∴ `ρ_1/"A"_1 = ρ_2/"A"_2`

`"A"_1/"A"_2 = ρ_1/ρ_2`

= `(2.63 xx 10^-8)/(1.72 xx 10^-8)`

= `2.63/1.72`

Mass of the aluminium wire,

m1 = Volume × Density

= A1l1 × d= A1 l1d1 ….......(3)

Mass of the copper wire,

m2 = Volume × Density

= A2l2 × d= A2 l2d2 ….........(4)

Dividing equation (3) by equation (4), we obtain

`"m"_1/"m"_2 = ("A"_1"l"_1"d"_1)/("A"_2"l"_2
"d"_2)`

for I1 = I2,

`"m"_1/"m"_2 = ("A"_1"d"_1)/("A"_2"d"_2)`

For `"A"_1/"A"_2 = 2.63/1.72`,

`"m"_1/"m"_2 = 2.63/1.72 xx 2.7/8.9` = 0.46

It can be inferred from this ratio that m1 is less 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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Chapter 3: Current Electricity - Exercise [Page 129]

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NCERT Physics Class 12
Chapter 3 Current Electricity
Exercise | Q 3.16 | Page 129
NCERT Physics Class 12
Chapter 3 Current Electricity
Exercise | Q 16 | Page 129

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