Two rectangular blocks, having identical dimensions, can be arranged either in configuration-I or in configuration-II as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2k$. The temperature difference between the ends along the $x$-axis is the same in both the configurations. It takes $9\,\text{s}$ to transport a certain amount of heat from the hot end to the cold end in the configuration-I. The time to transport the same amount of heat in the configuration-II is

Step 1: Relate heat transfer, thermal resistance, and time.

The heat transferred $Q$ through a thermal resistance $R_{\text{th}}$ under a temperature difference $\Delta T$ in time $t$ is given by the relation:

$$Q = \frac{\Delta T}{R_{\text{th}}} t$$

Since the heat $Q$ and the temperature difference $\Delta T$ are the same in both configurations, the time $t$ is directly proportional to the equivalent thermal resistance:

$$t \propto R_{\text{th}}$$

Step 2: Calculate the equivalent thermal resistance for Configuration-I.

In Configuration-I, the two blocks are connected in series along the direction of heat flow. Let each block have length $L$ and cross-sectional area $A$.

$$R_{\text{I}} = \frac{L}{kA} + \frac{L}{2kA}$$ $$R_{\text{I}} = \frac{3L}{2kA}$$

Step 3: Calculate the equivalent thermal resistance for Configuration-II.

In Configuration-II, the two blocks are connected in parallel along the direction of heat flow. The total thermal conductance $G_{\text{II}}$ is the sum of the individual conductances:

$$G_{\text{II}} = \frac{kA}{L} + \frac{2kA}{L}$$ $$G_{\text{II}} = \frac{3kA}{L}$$

The equivalent thermal resistance $R_{\text{II}}$ is the reciprocal of the total conductance:

$$R_{\text{II}} = \frac{1}{G_{\text{II}}} = \frac{L}{3kA}$$

Step 4: Determine the time required for Configuration-II.

Using the proportionality between time and thermal resistance:

$$\frac{t_{\text{II}}}{t_{\text{I}}} = \frac{R_{\text{II}}}{R_{\text{I}}}$$ $$\frac{t_{\text{II}}}{t_{\text{I}}} = \frac{\frac{L}{3kA}}{\frac{3L}{2kA}}$$ $$\frac{t_{\text{II}}}{t_{\text{I}}} = \frac{2}{9}$$

Substitute the given time $t_{\text{I}} = 9\,\text{s}$:

$$t_{\text{II}} = 9 \times \frac{2}{9}$$ $$t_{\text{II}} = 2\,\text{s}$$

Answer: $2\,\text{s}$