If K 1 and K 2 are the thermal conductivities L 1 and L 2 are the lengths and A 1 and A 2 are the cross sectional areas of steel and copper rods respectively such that K 2 K 1 = 9 , A 1 A 2 = 2 , L 1

Question

If K 1 and K 2 are the thermal conductivities L 1 and L 2 are the lengths and A 1 and A 2 are the cross sectional areas of steel and copper rods respectively such that K 2 K 1 = 9 , A 1 A 2 = 2 , L 1 L 2 = 2 . Then, for the arrangement as shown in the figure. The value of temperature T of the steel - copper junction in the steady state will be

TestHub · Accepted Answer

At steady state, the heat conducted by both the rods will be the same.
The rate of heat conduction is given by, $\frac{d Q}{d t} = \frac{K A}{l} ∆ T$ . Therefore, if the temperature of the junction is $T$ ,
$\frac{K_{1} A_{1}}{l_{1}} (T_{1} - T) = \frac{K_{2} A_{2}}{l_{2}} (T - T_{2})$
$\Rightarrow \frac{450 - T}{T - 0} = \frac{K_{2} A_{2} l_{1}}{K_{1} A_{1} l_{2}} = 9 \times \frac{1}{2} \times 2$
$\Rightarrow 450 - T = 9 T \Rightarrow T = 45 ° C$

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