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(a) Write the relation between resistance R\text R and electrical resistivity ρ\text ρ of the material of a conductor in the shape of cylinder of length l\text l and area of cross-section A\text A. Hence derive the SI unit of electrical resistivity.

(b) The resistance of a metal wire of length 3 m is 60 Ω. If the area of cross-section of the wire is 4 × 107 m2, calculate the electrical resistivity of the wire.

(c) State how would electrical resistivity be affected if the wire (of part 'b') is stretched so that its length is doubled. Justify your answer.

Current Electricity

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Answer

(a) Resistance of a cylindrical conductor is given by,

R=ρlA\text R = \text ρ \dfrac{\text l}{\text A}

Then,

SI unit of R=SI unit of ρ×SI unit of lSI unit of A⇒SI unit of ρ=SI unit of R×SI unit of ASI unit of l=Ω×m2m⇒SI unit of ρ=Ω.m\text {SI unit of R} = \text {SI unit of ρ} \times \dfrac{\text {SI unit of l}}{\text {SI unit of A}} \\[1em] \Rightarrow \text {SI unit of ρ} = \dfrac {\text {SI unit of R}\times \text {SI unit of A}}{\text {SI unit of l}} \\[1em] = \dfrac{\text Ω \times \text m^2 }{\text m} \\[1em] \Rightarrow \text {SI unit of ρ} = \text {Ω.m}

Hence, SI unit of ρ is Ω.m.

(b) Given,

  • Resistance of the wire (R\text R) = 60 Ω
  • Length of the wire (l\text l) = 3 m
  • Area of cross-section of the wire (A\text A) = 4 × 10-7 m2

As, resistance of a cylindrical conductor is given by,

R=ρlA\text R = \text ρ \dfrac{\text l}{\text A}

where ρ\text ρ is electrical resistivity of the wire.

Then,

⇒ρ=RAl=60×4×10−73=240×10−73⇒ρ=80×10−7 Ωm ⇒ρ=8×10−6 Ωm\Rightarrow \text ρ = \dfrac{\text {RA}}{\text l} \\[1em] = \dfrac{60\times 4 \times 10^{-7}}{3} \\[1em] = \dfrac{240\times 10^{-7}}{3} \\[1em] \Rightarrow \text ρ = 80\times 10^{-7} \text { Ωm} \\[1em]\ \Rightarrow \text ρ = 8\times 10^{-6} \text { Ωm} \\[1em]

Hence, the electrical resistivity of the wire is 8 × 10-6 Ωm.

(c) When the wire is stretched, its length increases and area of cross-section decreases, but resistivity (ρ) depends only on the nature of the material and temperature, not on its dimensions.

Hence, electrical resistivity remains unchanged if the material and temperature are the same, even though the resistance will increase.

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