Question

The resistance of a carbon filament at $0^{o}C$ is 104$\Omega$. It is connected in series with an iron wire. The temperature coefficients of resistivity of carbon and iron are $-0.0003$ and $0.0052$ per ${}^{o}C$ respectively. What must be the resistance of iron wire so that the combined resistance does not change with temperature?

• A. 1.8 $\Omega$
• B. 3.7 $\Omega$
• C. 6.0 $\Omega$
• D. 15 $\Omega$

Answer 1

The correct option is C 6.0 $\Omega$

Let, $R_c$ and $R_i$ be the resistances of copper and iron wires

respectively. ${\alpha }_c$ and ${\alpha }_i$ be the temperature coefficients of resistivity for copper and iron wires respectively.

The change in resistance with temperature is given by

$R=R_0+R_0\alpha \Delta T$

where, $\Delta T$ is the temperature difference. and $R_0=R_c+R_i$ at $0^{\circ }C$

the combined resistance does not change with temperature hence, the decrease in resistance of copper wire is equal to the increase in resistance of iron wire.

Therefore,

$-R_c{\alpha }_c\Delta T=R_i{\alpha }_i\Delta T$

$-104\times (-0.0003)=R_i\times 0.0052$

Negative sign indicates the decrement in resistance.

$0.0312=R_i\times 0.0052$

$R_i=6\Omega$