Question

In Ingen Hauz's experiment, the wax melts up to lengths 10 cm and 25 cm on two identical rods of different materials. Find the ratio of thermal conductivities of the two materials.
A. 1:6.25
B. 6.25:1
C. \[1:\sqrt {2.5} \]
D. 1:2.5

Answer 1

Hint:According to Ingen Hauz’s experiment if a number of identical rods made up of different metals are coated with wax and one of the ends is put in boiled water then a square of the length over which the wax melts is directly proportional to the thermal conductivity of the rod.

Formula Used:
According to Ingen Hauz's experiment the formula is,
\[\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{l_1}^2}}{{{l_2}^2}}\]
Where, \[{K_1},{K_2}\] are the thermal conductivity and \[{l_1},{l_2}\] are lengths of the rod.

Complete step by step solution:
In Ingen Hauz's experiment, the wax melts up to lengths of 10cm and 25cm on two identical rods of different materials. We need to find the ratio of thermal conductivities of the two materials.

As we know from Ingen Hauz’s experiment, the rate of heat transfer is constant. That is,
\[\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{l_1}^2}}{{{l_2}^2}}\]
Given, \[{l_1} = 10\,cm\] and \[{l_2} = 25\,cm\]
Substitute the value in the above equation we get,
\[\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{10}^2}{{25}^2} \\ \]
\[\Rightarrow \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{100}{625} \\ \]
\[\Rightarrow \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{1}{6.25} \\ \]
That is, \[{K_1}:{K_2} = 1:6.25\]
Therefore, the ratio of thermal conductivities of the two materials is 1:6.25.

Hence, option A is the correct answer.

Note: Here in the given problem it is important to remember the relation between the thermal conductivity of a metal rod is K and the length of the rod. In this experiment, several rods of different metals and of equal area of cross section and polish to ensure equal emissivity are coated with wax. One end of each of the rods is placed in an oil bath and heat is transferred from the end introduced into the bath towards the other end exposed outside causing the melting of wax on the surface. The thermal conductivities of different materials are compared and determined if that of any one material is known.