Question

Two identical rods of a metal are welded in series then 20 calories of heat flows through them in 4 minute. If the rods are welded in parallel then same amount of heat will flow in
A. 1 minute
B. 2 minute
C. 4 minute
D. 15 minute

Answer 1

Hint: In this question it is said that the two identical rods are connected in series then they are connected in parallel so we will find the heat flow in both the case and then we will find the time of heat flow by comparing both the equations.

Complete step by step answer:
Amount of heat flowing through them \[h = 20cal\]
Total time for which heat flows \[t = 4\min \]
Let the length of the rod be $l$
We know the heat flowing through a medium for a given time is given by the formula
\[\dfrac{{dQ}}{{dt}} = \dfrac{{KA\Delta T}}{l} - - (i)\]
Initially is said that two identical rods of a mental are welded in series hence the total length of the rod will be equal to \[ = 2l\] and 20 calorie of heat flows through it in 4 minute, hence we can write equation (i) as:
\[
  \dfrac{{dQ}}{{dt}} = \dfrac{{KA\Delta T}}{{2l}} \\
  \Rightarrow \dfrac{{20}}{4} = \dfrac{{KA\Delta T}}{{2l}} - - (ii) \\
 \]
Now it is said that the two rods are welded in parallel with the same amount of heat flowing through then,
So the area of the cross section will become 2A
Now we can write equation (i) as
\[\dfrac{{20}}{t} = \dfrac{{K\left( {2A} \right)\Delta T}}{l} - - (iii)\]
Now we divide equation (ii) by equation (iii), so we get
\[\dfrac{{\left( {\dfrac{{20}}{t}} \right)}}{{\left( {\dfrac{{20}}{4}} \right)}} = \dfrac{{\left( {\dfrac{{K\left( {2A} \right)\Delta T}}{l}} \right)}}{{\left( {\dfrac{{KA\Delta T}}{{2l}}} \right)}}\]
Hence by further solving we get
\begin{align*}
\dfrac{{20}}{t} \times \dfrac{4}{{20}} &= \dfrac{{K\left( {2A} \right)\Delta T}}{l} \times \dfrac{{2l}}{{KA\Delta T}} \\
 \Rightarrow \dfrac{4}{t} &= 4 \\
 \Rightarrow t &= \dfrac{4}{4} \\
 \therefore &1\min \\
 \end{align*}
Hence when the rods are welded in parallel then same amount of heat will flow in\[ = 1\min \]
Option A is correct.

Note:It is interesting to note here that if the metals are connected in series then the length of the welded metal will increase and thus, it takes more time for the heat to get transferred between the end points in comparison to the scenario when the metals are welded in parallel as the length will be reduced.