Question

Two rods having thermal conductivities in the ratio of 5:3 and having an equal length and equal cross-section are joined face to face. If the temperature of the free end of the first rod is \[{100^0}C\] and the temperature of the free end of the second rod is \[{20^0}C\], find the temperature of the junction.
A. \[{90^0}C\]
B. \[{85^0}C\]
C. \[{70^0}C\]
D. \[{50^0}C\]

Answer 1

Hint: In order to solve this problem we need to understand the rate of heat transfer. The rate of flow of heat is the amount of heat that is transferred per unit of time. Here, using the formula for heat flow we are going to find the solution. When the two rods are kept in contact with each other then, the temperature at the junction of 2 rods is known as junction temperature.

Formula Used:
To find the heat flow the formula is,
\[\dfrac{Q}{t} = KA\dfrac{{\Delta \theta }}{L}\]
Where,
A is cross-sectional area of rod
\[\Delta \theta \] is temperature difference between two ends of the rod
L is length of the rod
K is thermal conductivity

Complete step by step solution:
Consider two rods having thermal conductivities in the ratio of 5:3 and having an equal length and equal cross-section are joined face to face. If the temperature of the free end of the first rod is \[{100^0}C\]and the temperature of the free end of the second rod is \[{20^0}C\], we need to find the temperature of the junction.

The rate of flow of heat for the two rods is,
\[{\left( {\dfrac{Q}{t}} \right)_1} = {K_1}A\dfrac{{\Delta \theta }}{L}\]
And, \[{\left( {\dfrac{Q}{t}} \right)_2} = {K_2}A\dfrac{{\Delta \theta }}{L}\]
As we know that, at steady-state,
\[{\left( {\dfrac{Q}{t}} \right)_1} = {\left( {\dfrac{Q}{t}} \right)_2} \\ \]
\[\Rightarrow {K_1}A\dfrac{{\Delta \theta }}{L} = {K_2}A\dfrac{{\Delta \theta }}{L} \\ \]
\[\Rightarrow {K_1}\left( {{\theta _1} - \theta } \right) = {K_2}\left( {\theta - {\theta _2}} \right) \\ \]
Here, \[{K_1} = 5\], \[{K_2} = 3\], \[{\theta _1} = {100^0}C\]and \[{\theta _2} = {20^0}C\]

Then, above equation will become,
\[5\left( {{{100}^0} - \theta } \right) = 3\left( {\theta - {{20}^0}} \right) \\ \]
\[\Rightarrow {500^0} - 5\theta = 3\theta - {60^0} \\ \]
\[\Rightarrow 8\theta = {560^0} \\ \]
\[\Rightarrow \theta = \dfrac{{{{560}^0}}}{8} \\ \]
\[\therefore \theta = {70^0}\]
Therefore, the temperature of the common junction is, \[{70^0}\].

Hence, option C is the correct answer.

Note: Here, it is important to remember that when we are going to find the temperature of the junction, the steady state comes into picture. At steady state we equate the heat flow of both the rods, thereby calculating the temperature of the junction.