To solve this problem, we will use two important laws related to black body radiation: Wien's Displacement Law and Stefan-Boltzmann Law.$\\$1. Wien's Displacement Law:$\\ \lambda_{max} \cdot T = b \\$where $\lambda_{max}$ is the wavelength at which maximum energy is radiated, $T$ is the temperature, and $b$ is a constant.$\\$Initially, the black body radiates maximum energy at $\lambda_0,$ so:$\\ \lambda_0 \cdot T_1 = b \quad$(1)$\\$When the temperature changes, the maximum energy is radiated at $\frac{3}{4} \lambda_0,$ so:$\\ \frac{3}{4} \lambda_0 \cdot T_2 = b \quad$(2)$\\$From equations (1) and (2), we equate the constants:$\\ \lambda_0 \cdot T_1 = \frac{3}{4} \lambda_0 \cdot T_2 \\$Cancel $\lambda_0$ from both sides:$\\ T_1 = \frac{3}{4} T_2 \\$Rearrange to find $T_2: \\ T_2 = \frac{4}{3} T_1 \\$2. Stefan-Boltzmann Law:$\\ P = \sigma A T^4 \\$where $P$ is the power radiated, $\sigma$ is the Stefan-Boltzmann constant, $A$ is the surface area, and $T$ is the temperature.$\\$Initially, the power radiated is $P = \sigma A T_1^4. \\$When the temperature changes to $T_2,$ the power radiated becomes $nP: \\ nP = \sigma A T_2^4 \\$Substitute $T_2 = \frac{4}{3} T_1$ into the equation:$\\ nP = \sigma A \left(\frac{4}{3} T_1\right)^4 \\$Simplify:$\\ nP = \sigma A \left(\frac{256}{81} T_1^4\right) \\$Since $P = \sigma A T_1^4,$ substitute $P$ into the equation:$\\ nP = P \cdot \frac{256}{81} \\$Thus, $n = \frac{256}{81}. \\$Therefore, the value of $n$ is $\frac{256}{81},$ which corresponds to Option 1.