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 \[\alpha \] is temperature coefficient of resistance & Neglect linear expansion.
(A) \[\dfrac{{\alpha _c\rho _c}}{{\alpha _b\rho _b}}\]
(B) \[\dfrac{{\alpha _c\rho B}}{{\alpha _b\rho _c}}\]
(C) \[\dfrac{{\alpha _b\rho _c}}{{\alpha _c\rho _b}}\]
(D) \[\dfrac{{\alpha _b\rho _b}}{{\alpha _c\rho _c}}\]

Answer 1

Hint: First write the expression for the resistivity of the disc and then substitute the values of resistance, area and length of the in the expression. In this way the ratios of the thickness of the brass disc to that of the carbon disc is calculated.

Complete step by step answer:
The resistivity of a substance is the characteristic property of a substance that having edges of unit length, with the current flows normal to opposite faces and is distributed uniformly over them.
The electrical resistance per unit of cross-sectional area and per unit length at a particular temperature is termed as electrical resistivity. Ohm meter is the standard unit of the electrical resistance.
Write the expression for the resistivity.
\[\rho = R\dfrac{A}{l}\]
Here, \[\rho \] is the resistivity of the material in ohm meter, \[E\] is magnitude of electric field in volt per meter, and density in amperes per square meter. \[A\] is area of the cross section.
According to the question, it is given that the resistance of the cylinder is independent of the temperature.
Substitute the values,
\[\Rightarrow \dfrac{{\rho _bl_b}}{A}\left( {\alpha _b\Delta {{T}}} \right) = \dfrac{{\rho _cL_c}}{A}\left( {\alpha _c\Delta T} \right)\]
So, the ratio of thickness of the brass disc to that of the carbon disc is
\[\Rightarrow \dfrac{{{l_b}}}{{{l_c}}} = \dfrac{{\rho _c\alpha _c}}{{\rho _b\alpha _b}}\]
Therefore, the ratio of thickness of the brass disc to that of the carbon disc is \[\left( {{A}} \right)\], \[\dfrac{{\rho _c\alpha _c}}{{\rho _b\alpha _b}}\].

Note: As we know that the resistance of the cylinder is independent of temperature so do not consider it variable. The temperature does not change when other properties are changed.