TS EAMCET · Physics · Thermodynamics

Two rods whose lengths are $l_{1}$ and $l_{2}$ with heat conductivity co-efficients $k_{1}$ and $k_{2}$ are placed end to end. The heat conductivity coefficient of a uniform rod of length $l_{1} + l_{2}$ whose conductivity is same as that of the system of these two rods is

A $\frac{(l_{1} + l_{2}) k_{1} k_{2}}{k_{2} l_{1} + k_{1} l_{2}}$
B $\frac{(l_{1} + l_{2}) k_{1} k_{2}}{k_{1} l_{1} + k_{2} l_{2}}$
C $\frac{k_{1} l_{1} + k_{2} l_{2}}{(l_{1} + l_{2}) k_{1} k_{2}}$
D $\frac{k_{1} l_{2} + k_{2} l_{1}}{(l_{1} + l_{2}) k_{1} k_{2}}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{(l_{1} + l_{2}) k_{1} k_{2}}{k_{2} l_{1} + k_{1} l_{2}}$

Step-by-step Solution

Detailed explanation

When two rods are connected in series then the heat flow per unit time is the same across each rod, it means, H=H1=H2, ⇒kAT1-T2l1+l2=k1AT1-TJl1=k2ATJ-T2l2, first, calculate the junction temperature by the idea, H1=H2=k1AT1-TJl1=k2ATJ-T2l2⇒TJ=k1T1l1+k2T2l2k1l1+k2l2 it…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Two rods whose lengths are l1 and l2 with heat conductivity co-efficients k1 and k2 are placed end to end. The heat conductivity coefficient of a uniform rod of length l1+l2 whose conductivity is same as that of the system of these two rods is

Step-by-step Solution

Two rods whose lengths are $l_{1}$ and $l_{2}$ with heat conductivity co-efficients $k_{1}$ and $k_{2}$ are placed end to end. The heat conductivity coefficient of a uniform rod of length $l_{1} + l_{2}$ whose conductivity is same as that of the system of these two rods is