The peak wavelength of radiation emitted by a black body at a temperature of 2000 K is 1.45 µm. If the peak wavelength of emitted radiation changes to 2.90 µm, then the temperature (in K) of the black body is

The correct option is 1000.

To find the new temperature of the black body, we use Wien's Displacement Law. Wien's Displacement Law states that the wavelength corresponding to the peak of the emission spectrum (${\lambda }_{\text{max}}$) of a black body is inversely proportional to its absolute temperature ($T$). This relationship can be expressed mathematically as:
${\lambda }_{\text{max}}·T=\text{constant}$

From this relation, we can write the equation for two different states of the black body:
${\lambda }_1·T_1={\lambda }_2·T_2$

Let's identify the given values from the problem statement:
Initial temperature ($T_1$) = 2000 K
Initial peak wavelength (${\lambda }_1$) = 1.45 µm
New peak wavelength (${\lambda }_2$) = 2.90 µm

We need to find the new temperature ($T_2$). Rearranging the formula to solve for $T_2$ gives:
$T_2=\frac{{\lambda }_1·T_1}{{\lambda }_2}$

Substituting the given values into the equation:
$T_2=\frac{1.45\text{µm}·2000\text{K}}{2.90\text{µm}}$

We can simplify the ratio of the wavelengths:
$\frac{1.45}{2.90}=\frac{1}{2}$

Now, substitute this back to calculate $T_2$:
$T_2=\frac{1}{2}·2000\text{K}=1000\text{K}$

Therefore, the temperature of the black body is 1000 K.