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

To solve the problem of finding the resistances \( R_1 \) and \( R_2 \) of the carbon and iron resistors that together make a total resistance of \( 20 \, \Omega \) with a zero thermal coefficient of resistance, we can follow these steps:

### Step 1: Understand the given values
We know:
- The total resistance \( R_1 + R_2 = 20 \, \Omega \)
- The temperature coefficients of resistance:
- \( \alpha_{\text{carbon}} = -0.5 \times 10^{-3} \, ^\circ C^{-1} \)
- \( \alpha_{\text{iron}} = 5 \times 10^{-3} \, ^\circ C^{-1} \)
...