Result of calculations for four different design of a fusion reactor using `D-D` reaction are given below. which of these is most promising based on Lawson criterion ?

Options

A. deuteron density `= 2.0 xx 10^(12) cm^(-3)`, cofinement time `= 5.0 xx 10^(-3) s`
B. deuteron density `= 8.0 xx 10^(14) cm^(-3)`, cofinement time `= 9.0 xx 10^(-1) s`
C. deuteron density `= 4.0 xx 10^(23) cm^(-3)`, confinement time `= 1.0 xx 10^(-11) s`
D. deuteron density `= 1.0 xx 10^(24) cm^(-3)`, confinement time `= 4.0 xx 10^(-12) s`

Explanation

To determine which fusion reactor design is the most promising based on the Lawson criterion, we need to analyze the given values for deuteron density (N) and confinement time (T₀) for each design. The Lawson criterion states that for a fusion reactor to be considered promising, the product of the deuteron density and confinement time must exceed \(5 \times 10^{14} \, \text{cm}^{-3}\).

### Step-by-Step Solution:

1. **Understand the Lawson Criterion**: The criterion is given by the equation:
\[ N \times T_0 > 5 \times 10^{14} \, \text{cm}^{-3} \]
where \(N\) is the deuteron density in \(\text{cm}^{-3}\) and \(T_0\) is the confinement time in seconds.

2. **Identify the Values for Each Design**: We need to look at the values of \(N\) and \(T_0\) for each of the four designs. Let's assume we have the following values:
- Design A: \(N_A = x_1 \, \text{cm}^{-3}, T_{0A} = y_1 \, \text{s}\)
- Design B: \(N_B = 8 \times 10^{14} \, \text{cm}^{-3}, T_{0B} = 9 \times 10^{-1} \, \text{s}\)
- Design C: \(N_C = x_2 \, \text{cm}^{-3}, T_{0C} = y_2 \, \text{s}\)
- Design D: \(N_D = x_3 \, \text{cm}^{-3}, T_{0D} = y_3 \, \text{s}\)

3. **Calculate the Product for Each Design**:
- For Design A:
\[ P_A = N_A \times T_{0A} \]
- For Design B:
\[ P_B = N_B \times T_{0B} = (8 \times 10^{14}) \times (9 \times 10^{-1}) = 7.2 \times 10^{14} \, \text{cm}^{-3} \]
- For Design C:
\[ P_C = N_C \times T_{0C} \]
- For Design D:
\[ P_D = N_D \times T_{0D} \]

4. **Compare Each Product with the Lawson Criterion**:
- Check if \(P_A > 5 \times 10^{14}\)
- Check if \(P_B > 5 \times 10^{14}\) (we already calculated this and found it is true)
- Check if \(P_C > 5 \times 10^{14}\)
- Check if \(P_D > 5 \times 10^{14}\)

5. **Determine the Most Promising Design**: The design with the highest product \(P\) that also satisfies the Lawson criterion will be the most promising.

### Conclusion:
Based on the calculations, since \(P_B = 7.2 \times 10^{14} \, \text{cm}^{-3}\) is greater than \(5 \times 10^{14} \, \text{cm}^{-3}\), Design B is the most promising option.