A bar of gold and a bar of silver are used to conduct heat from one end maintained at 80.0oC and another end maintained at 30.0oC. If both bars have the same length L = 1.00m and the same cross sectional area A=100cm2 and are insulated from the outside, once steady state has been reached find the temperature of the interface. (Steady state occurs when heat flow is the same all throughout)
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To find the temperature of the interface between the two bars, we need to use the equation for steady-state heat transfer through a material:
Q = (kA∆T)/L
where:
Q is the rate of heat transfer (in this case, it is the same for both bars),
k is the thermal conductivity of the material,
A is the cross-sectional area of the bars,
∆T is the temperature difference between the ends of the bars, and
L is the length of the bars.
Since the bars are insulated from the outside, we can assume that the heat transfer occurs only between the two bars. Therefore, we can write the equation for both the gold and silver bars:
Q_gold = (k_gold * A * ∆T) / L
Q_silver = (k_silver * A * ∆T) / L
Since the rate of heat transfer is the same, we can set these equations equal to each other:
(k_gold * A * ∆T) / L = (k_silver * A * ∆T) / L
Since the cross-sectional area, length, and temperature difference are the same for both bars, we can cancel them out:
k_gold = k_silver
This means that the thermal conductivities of gold and silver are equal, which is a reasonable assumption for this problem.
Now, since k_gold = k_silver, the temperature at the interface between the two bars will be the average of the temperatures at the ends of the bars:
Temperature at the interface = (80.0oC + 30.0oC) / 2 = 55.0oC
Therefore, the temperature at the interface between the gold and silver bars is 55.0oC.