Originally Posted by **osalselaka** I am confused with this question, difficult to find steps to solve it.
A carbon cable and a tungsten cables are connected each and made a cable. The resistivity of tungsten is 5.6*10^-8Ωm and carbon is 3.5*10^-5Ωm. The temperature coefficient of resistivity ,tungsten=0.0045°C-1 and carbon=-0.0005°C-1. The resistance of resultant cable is not change with the temperature. **What is the ratio of the two lengths of the carbon and tungsten cables.**(Assume that the two cables have same cross sectional area). |

Pouillet's law is

$\displaystyle R = \rho \frac{l}{A}$

where R is resistance, $\displaystyle \rho$ is the resistivity, l is the length of the cable and A is the cross-sectional area of the cable. Furthermore, the resistivity can be temperature-dependent, which is usually expressed as a linear relationship relative to some reference resistivity, $\displaystyle \rho_0$, at a reference temperature, $\displaystyle T_0$.

$\displaystyle \rho = \rho_0 \left[1 + \alpha(T-T_0)\right]$

where T is temperature and $\displaystyle \alpha$ is the temperature coefficient of resistivity.

Using these laws we can build a relationship for the resistance of the whole cable. We know that two cables attached end to end add up in series, so

$\displaystyle R = R_c + R_t$

$\displaystyle \rho \frac{(l_t + l_c)}{A}= \rho_{0,t}\frac{l_t}{A} \left[1 + \alpha_t(T-T_0)\right] + \rho_{0,c}\frac{l_c}{A} \left[1 + \alpha_c(T-T_0)\right]$

$\displaystyle \rho (l_t + l_c)= \rho_{0,t} l_t \left[1 + \alpha_t(T-T_0)\right] + \rho_{0,c} l_c \left[1 + \alpha_c(T-T_0)\right]$

If we expand the brackets and then rearrange this so that all of the terms not dependent on T are on the LHS and all the ones that are are on the RHS, we get

$\displaystyle \rho (l_t + l_c) - \rho_{0,t} l_t + \rho_{0,t} l_t \alpha_t T_0 - \rho_{0,c} l_c + \rho_{0,c} l_c \alpha_c T_0 = \rho_{0,t} l_t \alpha_t T + \rho_{0,c} l_c \alpha_c T$

Since we know there is no temperature dependence, we can say that

$\displaystyle \rho_{0,t} l_t \alpha_t T + \rho_{0,c} l_c \alpha_c T = 0$

Which can occur because $\displaystyle \alpha_c$ is negative. that leaves

$\displaystyle \rho (l_t + l_c) - \rho_{0,t} l_t + \rho_{0,t} l_t \alpha_t T_0 - \rho_{0,c} l_c + \rho_{0,c} l_c \alpha_c T_0 = 0$

I think it should be fairly straightforward to rearrange this to get the ratio of the lengths of the cables.