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Standard XII

Physics

Temperature

Two conductin...

Question

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_{1} = 300 K$ and $T_{2} = 100 K$ as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_{1}$ and $K_{2}$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 K$ . Then $\frac{K_{1}}{K_{2}} =$

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Solution

In steady state, the heat flux across the two cylinders will be same .
$⎛ ⎜ ⎜ ⎜ ⎝ \frac{300 - 200}{\frac{L}{k_{1} π r^{2}}} ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{200 - 100}{\frac{L}{k_{2} π (2 r)^{2}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$

$\Rightarrow \frac{k_{1} π r^{2} \times 100}{L} = \frac{100 k_{2} π \times 4 r^{2}}{L}$

$\Rightarrow \frac{k_{1}}{k_{2}} = 4$

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In steady state, the heat flux across the two cylinders will be same . ⎛⎜ ⎜ ⎜⎝300−200Lk1πr2⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜⎝200−100Lk2π(2r)2⎞⎟ ⎟ ⎟ ⎟⎠ ⇒k1πr2×100L=100k2π×4r2L ⇒k1k2=4