Two rods one made of copper and other made of steel of the same length and same cross-sectional area are joined together. The thermal conductivity of copper and steel are 385Js−1K−1m−1) and 50Js−1K−1m−1, respectively. The free ends of copper and steel are held at 100°C and 0° C, respectively. The temperature at the junction is, nearly
Question
Two rods one made of copper and other made of steel of the same length and same cross-sectional area are joined together. The thermal conductivity of copper and steel are 385Js−1K−1m−1) and 50Js−1K−1m−1, respectively. The free ends of copper and steel are held at 100°C and 0° C, respectively. The temperature at the junction is, nearly
Solution
The temperature at the junction can be found using the formula for heat conduction, which is given by:
Q = kA(T1-T2)/d
where: Q is the heat transferred, k is the thermal conductivity, A is the cross-sectional area, T1 and T2 are the temperatures at the two ends, and d is the distance (or length of the rod).
Since the amount of heat transferred must be the same through both rods (as they are in series), we can set up the following equation:
k_copperA(T_junction - T_copper)/d_copper = k_steelA(T_steel - T_junction)/d_steel
Given that the lengths and cross-sectional areas of the rods are the same, we can simplify this to:
k_copper*(T_junction - T_copper) = k_steel*(T_steel - T_junction)
Substituting the given values:
385*(T_junction - 100) = 50*(0 - T_junction)
Solving this equation for T_junction gives:
T_junction = (385100 + 500) / (385 + 50) = 77.6°C
So, the temperature at the junction is approximately 77.6°C.
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