Question

The thermal conductivity of copper is four times that of brass. Two rods of copper and brass having the same length and cross-section are joined end to end. The free end of the copper is at 0°C and the free end of brass is at 100°C. The temperature of the junction is
A. 20°C
B. 40°C
C. 60°C
D. 10°C

Answer 1

Hint: This problem can be solved using the formula for thermal conductivity. Firstly, find the thermal conductivity of brass by substituting the values in the formula for thermal conductivity. Then, use the condition given in the question and find the thermal conductivity of the copper. Equate these two equations as their left-hand sides are equal. Then solve these equations and calculate the temperature at the junction.

Complete answer:

Thermal conductivity is given by,
$\dfrac {dQ}{dt} = \dfrac {KA \Delta T}{L}$
Where, K is the thermal conductivity
             Q is the amount of heat transferred
             A is the area of cross-section
             $\Delta T$ is the difference between the temperatures
             L is the length of the rod
Therefore, Thermal conductivity of brass can be given by,
$\dfrac {dQ}{dt} = \dfrac {KA (100-T)}{L}$ …(1}
Similarly, thermal conductivity of the copper can be given by,
$\dfrac {dQ}{dt} = \dfrac {4KA (T-0)}{L}$ …(2}
Equating equation. (1) and equation. (2) we get,
$\dfrac {KA (100-T)}{L}=\dfrac {4KA (T-0)}{L}$
$\Rightarrow 100-T = 4T$
$\Rightarrow 5T= 100$
$\Rightarrow T = 20°$
Hence, the temperature of the junction is 20°C.

So, the correct answer is option A i.e. 20°C.

Note:
Thermal conductivity of a solid or conductor is defined as the measure of the ability of the solid to conduct or transfer heat through it. It is an important property while deciding which metal is to be used for a specific purpose. Copper and Aluminium have the highest thermal conductivity. Its value is lowest for steel and bronze. Materials having low thermal conductivity are considered to be good insulating material. If the thickness of the material is doubled then the thermal conductivity becomes double.