Question

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 the middle has thermal conductivity K. The left end of the combination is maintained at temperature 3T and the right end the at T. The rods are thermally insulated from outside. In steady state, temperature at the left junction is T₁ and that the right junction is T₂. The ratio T₁/₂ T is:

विकल्प

• `3/2`

• `4/3`

• `5/3`

• `5/4`

Answer 1

`5/3`

Explanation:

In a steady state, the rate of flow of heat is the same through each rod.

`(ΔQ)/(Δt) = (2KA(3T - T_1))/l`

= `(KA(T_1 - T_2))/l`

= `(2KA(T_2 - T_1))/l`

2(3T − T₁) = (T₁ − T₂)

= 2(T₂ − T₁)

Considering, 2(3T − T₁) = (T₁ − T₂)

6T − 2T₁ = T₁ − T₂

6T = T₁ + 2T₁ − T₂

6T = 3T₁ − T₂ 

T = `(3T_1 - T_2)/6`

Considering, (T₁ − T₂) = 2(T₂ − T)

T₁ − T₂ = 2T₂ − 2T

2T = 2T₂ + T₂ − T₁

2T = 3T₂ − T₁

T = `(3T_2 - T_1)/2`

So, `(3T_1 - T_2)/6 = (3T_2 - T_1)/2`

⇒ 3T₁ − T₂ = 9T₂ − 3T₁

⇒ 6T₁ = 10T₂

⇒ `(T_1)/(T_2) = 10/6`

⇒ `(T_1)/(T_2) = 5/3`