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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 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 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

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 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

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Question & Answer (English)

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 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

  1. $\frac{3}{2}$
  2. $\frac{4}{3}$
  3. $\frac{5}{3}$ — Correct Answer
  4. $\frac{5}{4}$
Explanation:
Root Concept / Basics: Heat Conduction in Steady State. In a series combination, the rate of heat flow ($H = \frac{dQ}{dt}$) is constant through each part. The formula for heat current is $H = \frac{k A \Delta T}{L}$, where $k$ is thermal conductivity, $A$ is cross-sectional area, and $L$ is the length.

Correct Answer: $\frac{5}{3}$

Step I: Since the rods are in series, equate the heat currents: $H_1 = H_2 = H_3$.
$\frac{2KA(3T - T_1)}{L} = \frac{KA(T_1 - T_2)}{L} = \frac{2KA(T_2 - T)}{L}$.
Step II: Eliminate common terms $\frac{KA}{L}$ to simplify the equations: $2(3T - T_1) = (T_1 - T_2) = 2(T_2 - T)$.
Step III: Use the first two parts: $6T - 2T_1 = T_1 - T_2 \implies 3T_1 - T_2 = 6T$ (Equation 1).
Step IV: Use the second and third parts: $T_1 - T_2 = 2T_2 - 2T \implies T_1 - 3T_2 = -2T$ (Equation 2).
Step V: Multiply Equation 2 by 3: $3T_1 - 9T_2 = -6T$. Subtract this from Equation 1: $(3T_1 - T_2) - (3T_1 - 9T_2) = 6T - (-6T) \implies 8T_2 = 12T \implies T_2 = 1.5T$.
Step VI: Substitute $T_2$ into Equation 2 to find $T_1$: $T_1 - 3(1.5T) = -2T \implies T_1 - 4.5T = -2T \implies T_1 = 2.5T$.
Step VII: Calculate the ratio $\frac{T_1}{T_2} = \frac{2.5T}{1.5T} = \frac{25}{15} = \frac{5}{3}$.

प्रश्न एवं उत्तर (हिंदी)

तीन समान ऊष्मा चालक छड़ें श्रृंखला (series) में जुड़ी हुई हैं जैसा कि चित्र में दिखाया गया है। किनारों वाली छड़ों की ऊष्मीय चालकता 2K है जबकि बीच वाली छड़ की ऊष्मीय चालकता K है। संयोजन के बाएँ सिरे को 3T तापमान पर और दाएँ सिरे को T तापमान पर बनाए रखा जाता है। छड़ों को बाहर से ऊष्मीय रूप से अछूता (thermally insulated) रखा गया है। स्थिर अवस्था (steady state) में, बाएँ जंक्शन पर तापमान $T_{1}$ और दाएँ जंक्शन पर $T_{2}$ है। $T_{1}/T_{2}$ का अनुपात है

  1. $\frac{3}{2}$
  2. $\frac{4}{3}$
  3. $\frac{5}{3}$ — सही उत्तर
  4. $\frac{5}{4}$
स्पष्टीकरण:
मूल अवधारणा (Root Concept): स्थिर अवस्था में ऊष्मा चालन (Heat Conduction)। श्रृंखला संयोजन में, प्रत्येक भाग से ऊष्मा प्रवाह की दर ($H = \frac{dQ}{dt}$) स्थिर होती है। ऊष्मा धारा (heat current) का सूत्र $H = \frac{k A \Delta T}{L}$ है, जहाँ $k$ ऊष्मीय चालकता है, $A$ अनुप्रस्थ काट का क्षेत्रफल है, और $L$ लंबाई है।

सही उत्तर: $\frac{5}{3}$

Step I: चूँकि छड़ें श्रृंखला में हैं, ऊष्मा धाराओं को समान करें: $H_1 = H_2 = H_3$।
$\frac{2KA(3T - T_1)}{L} = \frac{KA(T_1 - T_2)}{L} = \frac{2KA(T_2 - T)}{L}$।
Step II: समीकरणों को सरल बनाने के लिए सामान्य पदों $\frac{KA}{L}$ को हटा दें: $2(3T - T_1) = (T_1 - T_2) = 2(T_2 - T)$।
Step III: पहले दो भागों का उपयोग करें: $6T - 2T_1 = T_1 - T_2 \implies 3T_1 - T_2 = 6T$ (समीकरण 1)।
Step IV: दूसरे और तीसरे भाग का उपयोग करें: $T_1 - T_2 = 2T_2 - 2T \implies T_1 - 3T_2 = -2T$ (समीकरण 2)।
Step V: समीकरण 2 को 3 से गुणा करें: $3T_1 - 9T_2 = -6T$। इसे समीकरण 1 से घटाएं: $(3T_1 - T_2) - (3T_1 - 9T_2) = 6T - (-6T) \implies 8T_2 = 12T \implies T_2 = 1.5T$।
Step VI: $T_1$ ज्ञात करने के लिए $T_2$ को समीकरण 2 में प्रतिस्थापित करें: $T_1 - 3(1.5T) = -2T \implies T_1 - 4.5T = -2T \implies T_1 = 2.5T$।
Step VII: अनुपात $\frac{T_1}{T_2} = \frac{2.5T}{1.5T} = \frac{25}{15} = \frac{5}{3}$ की गणना करें।

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