Question

Heat is flowing through two cylindrical rods made of same materials whose ends are maintained at similar temperatures. If diameters of the rods are in ratio $1:2$ and lengths in the ratio $2:1$, then the ratio of thermal current through in steady-state is:

• A. $1:8$
• B. $1:4$
• C. $1:6$
• D. $4:1$

Answer 1

The correct option is A $1:8$

The thermal current is nothing but the rate of flow heat,

$I=\frac{Q}{t}=K\Delta T\frac{A}{l}$. . . . . .(1)

Where,

$\Delta T=$ change in temperature

$A=\pi r^2$, cross-sectional area of the cylindrical rod.

$K=$ Thermal conductivity.

$l=$ length of the rod

From equation (1)

Thermal current is $I=4\pi K\Delta T\frac{d^2}{l}$

$I\alpha \frac{d^2}{l}$ where, $d=$ diameter of the cylindrical rod

Given,

$l_1:l_2=2:1$

$d_1:d_2=1:2$

The ratio of the thermal current is,

$\frac{I_1}{I_2}=(\frac{d_1}{d_2}{)}^2\times \frac{l_2}{l_1}$

$\frac{I_1}{I_2}=(\frac{1}{2}{)}^2\times \frac{1}{2}$

$\frac{I_1}{I_2}=\frac{1}{8}$

$I_1:I_2=1:8$

Option A is correct.