Question

A given copper wire has a resistance of 5.10 ? at 20.0°C while a tungsten wire of the same diameter has a resistance of 4.67 ? at 20.0°C. At what temperature will the two wires have the same resistance?

Answer 1

here,

dR / Ro = * alpha * dT

let c represent copper
let w represent tungsten
let R be the common resistance at some future temperature T

for copper

dRc / Roc = * alpha * cdT
(R - Roc) / Roc = * alpha * c(T - 20)

R - Roc = Roc * alpha * c(T - 20)
R = Roc * alpha * c(T - 20) + Roc

for tungsten
dRw / Row = * alpha * wdT
(R - Row) / Row = * alpha * w(T - 20)

substitute for R

(Roc * alpha * c(T - 20) + Roc - Row) / Row = * alpha * w(T - 20)

solve for T

Roc * alpha * c(T - 20) + Roc - Row = Row * alpha * w(T - 20)

Roc * alpha * cT - 20Roc * alpha * c + Roc - Row = Row * alpha * wT - 20Row * alpha * w

- 20Roc * alpha * c + Roc - Row + 20Row * alpha * w = Row * alpha * wT - Roc * alpha * cT

20(Row * alpha * w - Roc * alpha * c) + Roc - Row = T(Row * alpha * w - Roc * alpha * c)

T = [20(Row * alpha *
w - Roc * alpha * c) + Roc - Row] / (Row * alpha * w - Roc * alpha * c)

T = [20 * (4.67 * (4.50 * 10^-3)) - 5.1 * (3.90 * 10^-3 ) + 5.1 - 4.67] / (4.67 * (4.5 * 10^-3) - 5.1 * (3.90 * 10^-3))

T = 738.1 degree C

the temprature is 738.1 degree C