r₁ / r₂ = 1.41
The resistance, R, of a wire is related to its resistivity, ρ, its crossectional area, A, and its length, L, as follows;
R = ρL/A --------------------(i)
Where;
A = π x r² [r = radius of the wire]
Therefore, equation (i) can be re-written as;
R = ρL / (π x r²) --------------(ii)
(i) Now, For the first wire;
R = R
r = r₁
L = L
Substitute these values into equation (ii) as follows;
R = ρL / (π x r₁²)
Make r₁ subject of the formula;
r₁² = ρL / (π x R)
r₁ = √{ρL / (πR)]
(ii) Also, for the second wire;
R = 2R
r = r₂
L = L
Substitute these values into equation (ii) as follows;
2R = ρL / (π x r₂²)
Make r₂ subject of the formula;
r₂² = ρL / (π x 2R)
r₂ = √{ρL / (2πR)]
(iii) Now let's find the ratio of r₁ / r₂
r₁ = √{ρL / (πR)]
r₂ = √{ρL / (2πR)]
=> r₁ / r₂ = √ [ {ρL / (πR)] / {ρL / (2πR)] ]
=> r₁ / r₂ = √ [ {ρL / (πR)] ÷ {ρL / (2πR)] ]
=> r₁ / r₂ = √ [ {ρL / (πR)] x {2πR / (ρL)] ]
=> r₁ / r₂ = √ [ {2πR / (πR)] ]
=> r₁ / r₂ = √ [2]
=> r₁ / r₂ = √2
=> r₁ / r₂ = 1.41421
=> r₁ / r₂ = 1.41
Therefore, the ratio r₁ / r₂ = 1.41
