To solve this problem, we will use Wien's Displacement Law and the concept of black body radiation.$\\$Wien's Displacement Law is given by:$\\ \lambda_{max} \cdot T = b \\$where $\lambda_{max}$ is the wavelength at which the emission is maximum,$\\ T$ is the temperature, and $b$ is Wien's constant.$\\$Given:$\\ T = 5760 \,$K$\\ b = 2.88 \times 10^6 \,$nm K$\\$Calculate $\lambda_{max}: \\ \lambda_{max} = \frac{b}{T} = \frac{2.88 \times 10^6}{5760} \\ \lambda_{max} = 500 \,$nm$\\$This means the peak emission occurs at $500 \,$nm.$\\$Now, let's analyze the energy at different wavelengths:$\\$• At $250 \,$nm:$\\$This wavelength is shorter than $\lambda_{max},$ so the energy $U_1$ is less than the peak.$\\$• At $500 \,$nm:$\\$This is the peak wavelength, so the energy $U_2$ is maximum.$\\$• At $1000 \,$nm:$\\$This wavelength is longer than $\lambda_{max},$ so the energy $U_3$ is less than the peak.$\\$Comparing $U_1$ and $U_2: \\ U_2 > U_1 \\$Therefore, the correct option is:$\\$Option 2: $U_2 > U_1 \\$