Question

Two rods of same length and material are joined end to end. They transfer heat in 8 second. When they are joined in parallel they transfer same amount of heat in same conditions in time

• A. $3\text{ s}$
• B. $2\text{ s}$
• C. $1\text{ s}$
• D. $4\text{ s}$

Hint:

Memorize this useful network shortcut for competitive exams: for $n$ identical elements, the ratio of parallel to series effective resistance is always $\frac{R_p}{R_s} = \frac{1}{n^2}$. Since $n = 2$ here, the parallel combination is always $2^2 = 4$ times faster than the series configuration!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem presents two identical thermal rods transferring a fixed quantity of heat ($Q$) under uniform external temperature boundary differences. We are given that they take $t_s = 8\text{ seconds}$ to complete the heat transfer when arranged in a series configuration (end-to-end). We need to calculate the time $t_p$ required to transfer the exact same quantity of heat if they are reconfigured in a parallel arrangement.

Step 2: Key Formula or Approach:
Thermal conduction parameters map precisely to electrical current models. The heat transfer rate represents current, and the term $R = \frac{L}{KA}$ defines thermal resistance. The total heat energy equation is expressed as: $$Q = \frac{\Delta T \cdot t}{R_{\text{effective}}} \implies t \propto R_{\text{effective}}$$ Since heat quantity $Q$ and temperature drop $\Delta T$ are kept constant, the elapsed time is directly proportional to the total effective network resistance. For two identical resistances $R$ in series: $R_s = R + R = 2R$ For two identical resistances $R$ in parallel: $R_p = \frac{R \cdot R}{R + R} = \frac{R}{2}$

Step 3: Detailed Explanation:
Set up the direct proportionality ratio comparing both structural layouts: $$\frac{t_p}{t_s} = \frac{R_p}{R_s}$$ Substitute our derived structural equations for series and parallel resistances into this ratio: $$\frac{t_p}{t_s} = \frac{\left(\frac{R}{2}\right)}{2R} = \frac{1}{4}$$ This shows that changing the connection from series to parallel cuts the total effective resistance—and thus the time required—to exactly one-fourth of its original value: $$t_p = \frac{t_s}{4}$$ Substitute the given series transfer timeline parameter ($t_s = 8\text{ s}$) into the equation: $$t_p = \frac{8\text{ s}}{4} = 2\text{ seconds}$$

Step 4: Final Answer:
The time taken to transfer the same amount of heat in a parallel configuration is $2\text{ s}$, matching option (B).