Question

On heating one end of a rod, when the temperature of the whole rod will be uniform?
A. \[K = 1\]
B. \[K = 0\]
C. \[K = 100\]
D. \[K = \infty \]

Answer 1

Hint: In order to solve this problem we need to understand the thermal conductivity. The rate at which heat is transferred by conduction through a unit cross-section area of a material is known as thermal conductivity.

Formula used:
To find the flow of heat the formula is,
\[Q = KA\dfrac{{\Delta T}}{L}\] ……… (1)
Where,
A is the cross-sectional area, \[\Delta T\] is the temperature difference between two ends, L is the length of the cylinder and K is thermal conductivity. Here, we need to find the value of K.

Complete step by step solution:
Consider a uniform rod where the temperature will be the same at all the points. Suppose the temperature at every point is T which is uniform and, in this condition, we can say that \[\Delta T = 0\] and at each point the temperature is constant. From equation (1) K can be written as,
\[K = \dfrac{{QL}}{{A\Delta T}}\]
Here, we know that Q, L and A are constants
Therefore, we can write as,
\[K = \left( {\dfrac{{QL}}{A}} \right)\dfrac{1}{{\Delta T}}\]
\[\Rightarrow K \propto \dfrac{1}{{\Delta T}}\]

As we said that, \[\Delta T = 0\] then,
\[K = \infty \]
That is, if the temperature of the whole rod is the same, it means that heat is conducted from one end to the other infinitely fast. This is possible only when the thermal conductivity of the rod is infinite. Therefore, the temperature of the whole rod will be uniform when \[K = \infty \].

Hence, option D is the correct answer.

Note:The materials which have a high thermal conductivity are used in heat sinks whereas materials with low values are used as thermal insulators. In order to measure the thermal conductivities of materials there exist several methods, which are broadly classified into two types of techniques one is transient and other is steady-state techniques.