Question

If the ratio of the coefficient of thermal conductivity of silver and copper is 10:9, then find the ratio of the lengths up to which it will melt in the Ingen Hausz experiment.
A. 6:10
B. \[\sqrt {10} :3\]
C. 100:81
D. 81:100

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
\[K \propto {l^2}\]
Where, L is the length of the rod.

Complete step by step solution:
Consider two metals silver and copper which have the coefficient of thermal conductivity in the ratio of 10:9. We need to find the ratio of lengths over which wax will melt in the Ingen Hausz experiment. If K is the thermal conductivity and l is the length over which the wax melts then, we can write,
\[K \propto {l^2}\]
That is,
 \[\dfrac{{{K_1}}}{{{K_2}}} = {\left( {\dfrac{{{l_1}}}{{{l_2}}}} \right)^2}\]
\[\Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = \sqrt {\dfrac{{{K_1}}}{{{K_2}}}} \]

Substitute the value of \[{K_1} = 10\]and \[{K_1} = 9\]then the above equation will become,
\[\dfrac{{{l_1}}}{{{l_2}}} = \sqrt {\dfrac{{10}}{9}} \]
\[\Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{\sqrt {10} }}{3}\]
That is,
\[\therefore {l_1}:{l_2} = \sqrt {10} :3\]
Therefore, the ratio of the lengths up to which it will melt in the Ingen Hausz experiment is \[\sqrt {10} :3\].

Hence, option B is the correct answer.

Note: Here in the given question it is important to remember the statement of Ingen Hauz’s experiment. So, the statement says that thermal conductivity is directly proportional to the lengths of a given material. So, by calculating the ratio of thermal conductivity the ratio of lengths can be determined.