A black body has maximum wavelength ' $\lambda_{\mathrm{m}}$ ' at temperature 2000 K . Its maximum…

Wien's displacement law states that $\lambda_m T = b$, where $b$ is the Wien displacement constant, $\lambda_m$ is the peak wavelength, and $T$ is the absolute temperature. Consequently, $\lambda_m \propto \frac{1}{T}$.

For temperatures $T_1 = 2000$ K and $T_2 = 3000$ K, the corresponding peak wavelengths $\lambda_{m1}$ and $\lambda_{m2}$ satisfy $\lambda_{m1} T_1 = \lambda_{m2} T_2$.

Substituting the given $\lambda_{m1} = \lambda_m$ yields $\lambda_m \cdot 2000 = \lambda_{m2} \cdot 3000$, from which $\lambda_{m2} = \lambda_m \cdot \frac{2000}{3000} = \frac{2}{3} \lambda_m$.

The maximum wavelength at $3000$ K is $\frac{2}{3} \lambda_m$, corresponding to option D.