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Two conductors of the same material have their diameters in the ratio $1: 2$ and their lengths in the ratio $2: 1$. If the temperature difference between their ends is the same, then the ratio of amounts of heat conducted per second through them will be
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The correct answer is:
$1: 8$
The rate of heat conduction is given by
$$
H=\frac{d \theta}{d t}=\frac{k A}{l} \Delta T
$$
Here, $\Delta T$ is same for both conductors, so ratio of heat conduction,
$\begin{aligned} \frac{H_{1}}{H_{2}} &=\frac{A_{1}}{l_{1}} \times \frac{l_{2}}{A_{2}} \\ &=\frac{d_{1}^{2}}{l_{1}} \times \frac{l_{2}}{d_{2}^{2}} \quad\left(\because A=\frac{\pi d^{2}}{4}\right) \\ &=\left(\frac{d_{1}}{d_{2}}\right)^{2} \cdot \frac{l_{2}}{l_{1}} \\ &=\left(\frac{1}{2}\right)^{2} \times \frac{1}{2}=\frac{1}{8} \end{aligned}$
$$
H=\frac{d \theta}{d t}=\frac{k A}{l} \Delta T
$$
Here, $\Delta T$ is same for both conductors, so ratio of heat conduction,
$\begin{aligned} \frac{H_{1}}{H_{2}} &=\frac{A_{1}}{l_{1}} \times \frac{l_{2}}{A_{2}} \\ &=\frac{d_{1}^{2}}{l_{1}} \times \frac{l_{2}}{d_{2}^{2}} \quad\left(\because A=\frac{\pi d^{2}}{4}\right) \\ &=\left(\frac{d_{1}}{d_{2}}\right)^{2} \cdot \frac{l_{2}}{l_{1}} \\ &=\left(\frac{1}{2}\right)^{2} \times \frac{1}{2}=\frac{1}{8} \end{aligned}$
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