Question

Four rods of identical cross- sectional area and made from the same metal form the sides of the square. The temperature of two diagonally opposite points T and 2T respectively in the steady- state. Assuming that only heat conduction takes place, what will be the temperature difference between two points?
A. $\dfrac{\sqrt{2}+1}{2}T$
B. $\dfrac{2}{\sqrt{2}+1}T$
C. 0
D. None of these

Answer 1

Hint: We have been given that the temperature of two diagonally opposite points are T and 2T and are in steady state. And hence, we have to determine the temperature difference between the other two points. For that we have to assume the temperatures of B and D as \[{\theta _1}\] and \[{\theta _2}\] we should apply the concept \[\dfrac{{T - {\theta _1}}}{R} = \dfrac{{{\theta _1} - 2T}}{R}\] to determine the desired solution.

Formula used:
Temperature difference = Final temperature – Initial temperature

Complete step by step solution:
We are provided in the question that the temperature of two diagonally opposite points are \[{\rm{T}}\] and \[2\;{\rm{T}}\] respectively in the steady-state. And we are asked to determine the temperature difference between the other two points. Now, let us consider the temperature of B and D as \[{\theta _1}\] and \[{\theta _2}\]. We have already known that the heat flows through all the rods in the same amount. As there is no external heat source, we can write it as,
\[H = \dfrac{{\Delta T}}{R}\]
\[\Rightarrow R = \dfrac{L}{{{K^A}}}{\rm{ (same\,for\,all\,rods)}}\]
For \[\dfrac{{T - {\theta _1}}}{R} = \dfrac{{{\theta _1} - 2T}}{R}\]

Now, on cancelling the similar terms on both sides, we get
\[T - {\theta _1} = {\theta _1} - 2T\]
Now, we have to group the similar terms, we obtain
\[2{\theta _1} = T + 2T\]
Now, we have to simplify the right side of the equation, and solve for \[{\theta _1}\]
Therefore, we get
\[{\theta _1} = \dfrac{{3T}}{2}\]
Now, on solving for
\[\dfrac{{2T - {\theta _2}}}{R} = \dfrac{{{\theta _2} - T}}{R}\]
Now, on cancelling the similar terms, we have
\[2T - {\theta _2} = {\theta _2} - T\]

Now, we have to group the similar terms, we obtain
\[2{\theta _2} = 3T\]
Now, we have to simplify the right side of the equation, and solve for \[{\theta _2}\]. Therefore, we get
\[{\theta _2} = \dfrac{{3T}}{2}\]
The temperature difference will be
\[{\theta _1} - {\theta _2} = 0\]

Hence, the option C is correct.

Note: We could only use this conductivity formula because the flow is constant and the heat is only transferred via conduction. It is required to suppose that point C's temperature is between the temperatures of the other two places. The rate of heat flow would not be constant otherwise. So, students should keep in mind that heat flows through the rod in the same amount and hence applying the wrong formula will lead to the wrong solution.