It is desired to make a `20Omega` coil of wire, which has a zero thermal coefficient of resistance. To do this, a carbon resitor fo resistane `R_1` is placed in series with an iron resistor of resistance `R_2` the proportions of iron and carbon are so chosen that `R_1 +R_2 = 20Omega` for all temperatures near `20^@C.` How long are `R_1 and R_2`? `alpha_(carbon) =- 0.5xx10^(-3).^@C^(-1), alpha_(iron) = 5xx10^(-3) .^@C(-1)`

Explanation

We need `R-1(1+alpha_1Deltat) +R_2(1 + alpha_2Deltat) = 20. Because R_1 + R_2`
`= 20 "when" Delta t= 0,` we must have `R_1alpha_1 = -R_2alpha_2 =-"with" alpha_1 = -0.5`
`xx10^(-3) "and" alpha_2 = 5xx10^(-3).` Solving the two equations `R_1 +R_2`
`=20" and " R_1 = 10R_2` simultaneously leads to `R_1 = 18.18Omega` and
`R_2 = 1.82Omega`