Question

A brass disc and a carbon disc of same radius are assembled alternatively to make a cylindrical conductor. The resistance of the cylinder is independent of the temperature. The ratio of thickness of the brass disc to that of the carbon disc is
[${\alpha }_B$ and ${\alpha }_C$ are respectively the temperature coefficients of resistance of brass and carbon & ${\rho }_B$ and ${\rho }_C$ are respectively the resistivities of brass and carbon. Neglect linear expansion]

• A. $\left|\frac{{\alpha }_C{\rho }_C}{{\alpha }_B{\rho }_B}\right|$
• B. $\left|\frac{{\alpha }_C{\rho }_B}{{\alpha }_B{\rho }_C}\right|$
• C. $\left|\frac{{\alpha }_B{\rho }_C}{{\alpha }_C{\rho }_B}\right|$
• D. $\left|\frac{{\alpha }_B{\rho }_B}{{\alpha }_C{\rho }_C}\right|$

Answer 1

The correct option is A $\left|\frac{{\alpha }_C{\rho }_C}{{\alpha }_B{\rho }_B}\right|$

Variation of resistance with temperature is $R_1=\frac{{\rho }_B{l}_B}{A}(1+{\alpha }_B\Delta T)$
$R_2=\frac{{\rho }_C{l}_C}{A}(1+{\alpha }_C\Delta T)$

Here $l_B,l_C$ are thickness of brass and copper discs respectively.

For series connection of brass and copper discs,
$R_{equivalent}=R_1+R_2$
$R_{equivalent}=\frac{{\rho }_B{l}_B}{A}+\frac{{\rho }_B{l}_B}{A}{\alpha }_B\Delta T+\frac{{\rho }_C{l}_C}{A}+\frac{{\rho }_C{l}_C}{A}{\alpha }_C\Delta T$
Net resistance is independent of temperature,
$\Rightarrow \frac{{\rho }_B{l}_B{\alpha }_B\Delta T}{A}+\frac{{\rho }_C{l}_C{\alpha }_C\Delta T}{A}=0$
$\Rightarrow \frac{l_B}{l_C}=\left|\frac{{\alpha }_C{\rho }_C}{{\rho }_B{\alpha }_B}\right|$