Two identical plates P and Q, radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures $T_{P}$ and $T_{Q},$ respectively, with $T_{Q} < T_{P},$ as shown in Fig. 1. The radiated power transferred per unit area from P to Q is $W_{0}.$ Subsequently, two more plates, identical to P and Q, are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is $W_{S},$ then the ratio $W_{0}/W_{S}$ is ____.

Step 1: Define the initial state (Figure 1).

According to Stefan's law, the net radiated power per unit area $W_0$ transferred from plate P to plate Q is the difference between their emitted radiations.

$$W_0 = \sigma\left(T_P^{4} - T_Q^{4}\right)$$

Step 2: Set up the equations for the steady state with inserted plates (Figure 2).

Let the two inserted identical plates be A and B, with steady-state temperatures $T_A$ and $T_B$, placed in the order P, A, B, Q. In the steady state, the net power transferred per unit area $W_S$ must be equal across all three adjacent gaps.

$$W_S = \sigma\left(T_P^{4} - T_A^{4}\right)$$ $$W_S = \sigma\left(T_A^{4} - T_B^{4}\right)$$ $$W_S = \sigma\left(T_B^{4} - T_Q^{4}\right)$$

Step 3: Relate the steady-state power $W_S$ to the initial power $W_0$.

Sum the three equations to eliminate the intermediate temperatures $T_A$ and $T_B$.

$$3W_S = \sigma\left(T_P^{4} - T_Q^{4}\right)$$

Substitute the expression for $W_0$ from the first step.

$$3W_S = W_0$$

Step 4: Find the ratio of the powers.

$$\frac{W_0}{W_S} = 3$$

Answer: 3