Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity 2K while that in - Physics MCQ Physics | ExamTest.live | ExamTest.live

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Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity 2 K while that in the middle has thermal conductivity K . The left end of the combination is maintained at temperature 3 T and the right end at T . The rods are thermally insulated from outside. In steady state, temperature at the left junction is T_{1} and that at the right junction is T_{2} . The ratio T_{1} / T_{2} is:

A $3/2$
B $4/3$
C $5/3$
D $5/4$

Solution & Step-by-step Explanation

In steady state, heat current H is the same through all rods. H = \frac{Δ T}{R _{t h}}, where R_{t h} = \frac{L}{K A} .Since dimensions are identical, R \propto \frac{1}{K} .Let R_{0} be the resistance of the middle rod (K). Then resistance of side rods is R_{0} /2 . H = \frac{3 T - T _{1}}{R _{0} /2} = \frac{T _{1} - T _{2}}{R _{0}} = \frac{T _{2} - T}{R _{0} /2} .Equating 1st and 3rd: 2 (3 T - T_{1}) = 2 (T_{2} - T) ⟹ 3 T - T_{1} = T_{2} - T ⟹ T_{1} + T_{2} = 4 T .Equating 2nd and 3rd: T_{1} - T_{2} = 2 (T_{2} - T) ⟹ T_{1} - T_{2} = 2 T_{2} - 2 T ⟹ T_{1} - 3 T_{2} = - 2 T .We have: T_{1} + T_{2} = 4 T T_{1} - 3 T_{2} = - 2 T Subtracting (2) from (1): 4 T_{2} = 6 T ⟹ T_{2} = 1.5 T = \frac{3}{2} T .Substituting into (1): T_{1} = 4 T - 1.5 T = 2.5 T = \frac{5}{2} T .Ratio \frac{T _{1}}{T _{2}} = \frac{2.5 T}{1.5 T} = \frac{5}{3} .

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Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity 2 K while that in the middle has thermal conductivity K . The left end of the combination is maintained at temperature 3 T and the right end at T . The rods are thermally insulated from outside. In steady state, temperature at the left junction is T_{1} and that at the right junction is T_{2} . The ratio T_{1} / T_{2} is:

A

3/2

B

4/3

C

5/3

D

5/4

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Heat Transfer Thermal Conductivity Steady State

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