Question

What equal length of an iron wire and a copper-nickel alloy wire, each of 2 mm diameter connected parallel to give an equivalent resistance of 3 Ω?

(Given resistivities of iron and copper-nickel alloy wire are 12 µΩ cm and 51 µΩ cm respectively).

Options

• 82 m

• 110 m

• 97 m

• 90 m

Answer 1

97 m

Explanation:

Given that the dimensions of both wires are the same.

i.e., l_Cu-Ni = l_Fe = l and A_Fe = A_Cu-Ni = πr² 

From the relation, R = `rhol/A`, ρ-Resistivity.

Given ρ_Fe = 12 µΩ cm, ρ_Cu-Ni = 15 µΩ cm, R_eq. = 3Ω

`1/R_{eq} = 1/R_{Fe} + 1/R_{Cu-Ni}`

⇒ `1/3 = A/(rho_{Fe}l) + A/(rho_{CuNi}l)`

`l = 3A(1/(rho_{Fe}) + 1/(rho_{Cu-Ni}))`

= `3pir^2((rho_{Fe} + rho_{Cu-Ni})/(rho_{Fe} xx rho_{Cu-Ni}))`

`l = 3pi xx (1 xx 10^-3)^2[(12 + 51)/(12 xx 51)]`

= 97 m