Question

The figure below shows three cylindrical copper conductors along with their face areas and lengths compare the resistance and the resistivity of the three conductors justify your answer

Answer 1

TH required formula is R = ρL / A

where R = resistance through the material ; ρ = resistivity of the material ; L = length of the material ; A = cross sectional are of it.

PART 1(resistivity of all the 3 cylinders are same) :

For image (a); R₁ = ρ.L / A

For image (b); R₂ = ρ.(3L) / (A/3) = ρ.9L / A

For image (c); R₃ = ρ.(L/3) / (3.A) = ρ.L / 9.A

Therefore, R₁ : R₂ : R₃ = (ρ.L / A) : (ρ.9L / A) : (ρ.L / 9.A) = 1 : 9 :(1/9)

or, R₁ : R₂ : R₃ = 9 : 81 : 1

PART 2(in case those were made of different material, and resistance of all 3 are same) :

For image (a); R = ρ₁.L / A

or, ρ₁ = R.A / L

For image (b); R = ρ₂.(3L) / (A/3) = ρ.9L / A

or, ρ₂ = R.A / 9.L

For image (c); R = ρ₃.(L/3) / (3.A) = ρ₃.L / 9.A

or, ρ₃ = 9.R.A / L

Therefore, ρ₁ : ρ₂ : ρ₃ = (R.A / L) : (R.A / 9.L) : (9.R.A / L)

or, ρ₁ : ρ₂ : ρ₃ = 1 : (1/9) : 9 = 9 : 1 : 81

Answer 2

Answer:

PART 1(resistivity of all the 3 cylinders are same) :

For image (a); R₁ = ρ.L / A

For image (b); R₂ = ρ.(3L) / (A/3) = ρ.9L / A

For image (c); R₃ = ρ.(L/3) / (3.A) = ρ.L / 9.A

Therefore, R₁ : R₂ : R₃ = (ρ.L / A) : (ρ.9L / A) : (ρ.L / 9.A) = 1 : 9 :(1/9)

or, R₁ : R₂ : R₃ = 9 : 81 : 1

PART 2(in case those were made of different material, and resistance of all 3 are same) :

For image (a); R = ρ₁.L / A

or, ρ₁ = R.A / L

For image (b); R = ρ₂.(3L) / (A/3) = ρ.9L / A

or, ρ₂ = R.A / 9.L

For image (c); R = ρ₃.(L/3) / (3.A) = ρ₃.L / 9.A

or, ρ₃ = 9.R.A / L

Therefore, ρ₁ : ρ₂ : ρ₃ = (R.A / L) : (R.A / 9.L) : (9.R.A / L)

or, ρ₁ : ρ₂ : ρ₃ = 1 : (1/9) : 9 = 9 : 1 : 81