Three rods of identical cross-sectional area and made from the same metal from the sides of an isosceles triangle ABC, right angled at B. The points A and B are maintained at temperatures T and (√ 2)T respectively. In the steady state, the temperature of the point C is Tc. Assuming that only heat conduction takes place, Tc/T is

Question

Three rods of identical cross-sectional area and made from the same metal from the sides of an isosceles triangle ABC, right angled at B. The points A and B are maintained at temperatures T and (√ 2)T respectively. In the steady state, the temperature of the point C is Tc. Assuming that only heat conduction takes place, Tc/T is (A) (1/2(√ 2 -1)) (B) (3/(√ 2 +1)) (C) (1/√ 3(√ 2 -1)) (D) (1/(√ 2 +1))

Tardigrade Admin · Accepted Answer

The diagramatic representation of the given problem is shown in figure. Since, T

_{B} > T_{A}

the heat will flow from B to
A. Similarly, heat will also flow from B to C and C to A.
Applying the conduction formula

\frac{Δ Q}{Δ t} = \frac{K A}{l} (Δ T)

To the sides CA and BC, we get

(\frac{Δ T}{2 a})_{C A} = (\frac{Δ T}{a})_{BC} \Rightarrow \frac{T _{C} - T}{2 a} = \frac{2 T - T _{C}}{a}

3 T = T_{C} (2 + 1) \Rightarrow \frac{T _{C}}{T} = \frac{3}{( 2 + 1 )}

$\frac{1}{2 ( 2 - 1 )}$

$\frac{3}{( 2 + 1 )}$

$\frac{1}{3 ( 2 - 1 )}$

$\frac{1}{( 2 + 1 )}$

2(2​−1)1​

(2​+1)3​

3​(2​−1)1​

(2​+1)1​

$\frac{1}{2 ( 2 - 1 )}$

$\frac{3}{( 2 + 1 )}$

$\frac{1}{3 ( 2 - 1 )}$

$\frac{1}{( 2 + 1 )}$