Conduction Law: Differential Form

Q.

Two different metal rods of the same length have their ends kept at the same temperatures ${\theta }_1$ and ${\theta }_2$ with ${\theta }_2$ > ${\theta }_1$. If $A_1$ and $A_2$ are their cross-sectional areas and $k_1$ and $k_2$ their thermal conductivities, the rate of heat flow in the two rods will be same if

1. $\frac{k_1}{k_2}=\frac{A_1}{A_2}$

2. $\frac{k_1}{k_2}=\frac{A_2}{A_1}$

3. $\frac{k_1}{k_2}=\frac{A_1{\theta }_2}{A_2{\theta }_1}$

4. $\frac{k_1}{k_2}=\frac{A_1{\theta }_1}{A_2{\theta }_2}$

Q. A bar measured with a Vernier caliper is found to be $180mm$ long. The temperature during the measurement is ${10}^{\circ }C$. The measurement error will be if the scale of the Vernier caliper has been graduated at a temperature of ${20}^{\circ }C:(\alpha =1.2\times {10}^{-5}{}^{\circ }{C}^{-1}$. Assume that the length of the bar does not change).

1. $1.98\times {10}^{-1}mm$
2. $1.98\times {10}^{-3}mm$
3. $1.98\times {10}^{-2}mm$
4. $1.98\times {10}^{-4}mm$

Q.

The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively, are $T_2$ and $T_1$ ($T_2>T_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{A(T_2-T_1)K}{x}\right)f$, with f equal to

1. 1

2. $\frac{1}{2}$

3. $\frac{2}{3}$

4. $\frac{1}{3}$