To measure the effective thermal conductivity of an insulating material a spherical shell of inner radius 26 cm and outer radius 34 cm was constructed and a 100 w electrical bulb placed inside the centre. At steady state temperatures of inner and outer surfaces are measured to be 339 K and 311 K respectively. The thermal conductivity of material is

0.829 \frac{w}{m K}

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0.257 \frac{w}{m K}

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1.22 \frac{w}{m K}

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1.59 \frac{w}{m K}

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Solution

The correct option is B $0.257 \frac{w}{m K}$
$\begin{matrix} r_{1} & = 0.26 m r_{2} & = 0.34 m Q & = 100 W T_{1} & = 339 K T_{2} & = 311 K Q & = \frac{T_{1} - T_{2}}{\frac{r_{2} - r_{1}}{4 π K r_{1} r_{2}}} \Rightarrow 100 = \frac{339 - 311}{\frac{0.34 - 0.26}{4 π k (0.26) (0.34)}} \Rightarrow k & = 0.257 \frac{W}{m K} \end{matrix}$

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