Question

Two rods of the same material have diameters in the ratio of 1:2 and lengths in the ratio of 2:1. If the temperature difference between their ends is the same, then the ratio of heat conducted by them in a given time will be:
(a) 1:4
(b) 1:8
(c) 4:1
(d) 8:1

Answer 1

Hint: To solve this question one must have a concept about the topic thermal concept of matter where conduction takes place, to solve this question we will simply use the formula of heat flow for both the rods and compare them to get the required solution. Thermal conduction is the transfer of internal energy through the movement of electrons and tiny particle collisions within a substance. When molecules, atoms, and electrons collide, they combine to form a disorderly tiny kinetic and potential energy source known as internal energy.

Formula used: $Q = KA\dfrac{{dT}}{{dx}} = KA\dfrac{{\Delta T}}{l}$
Where, $Q$ is local heat flux density.
$K\,$ is material’s conductivity.
$\Delta T$ is the temperature gradient.
$l$ is the length of the rod.

Complete answer:
According to the question it is given that the ratio of the diameter of both the rods is 1:2 or we can say that $A \propto {D^2}$ means the ratio of their area is 1:4.
Also, 2:1 is the ratio of both the rods in terms of length.

Let, dimension of rod 1 be:
Length is 2l, therefore the area will be A.

Let, dimension of rod 2 be:
Length is l, therefore the area will be 4A.

Therefore,
${Q_1} = KA\dfrac{{\Delta T}}{{2l}}$ and
${Q_2} = 4KA\dfrac{{\Delta T}}{l}$

From here we can see the ratio of both the rods i.e., $\dfrac{{{Q_1} = KA\dfrac{{\Delta T}}{{2l}}}}{{{Q_1} = 4KA\dfrac{{\Delta T}}{l}}} = \dfrac{1}{8}$
Hence, the ratio of heat conducted by both the rods is 1:8.

Hence, the correct option is (b).

Note: An intriguing source of energy is heat. In addition to helping us prepare food, make ourselves comfortable, and preserve life, knowledge of its qualities is essential to numerous scientific domains. For instance, everything from creating heaters and comprehending seasonal variation to launching ships into space depends on understanding how heat is transported and the degree to which certain materials can interchange thermal energy.