A black body has maximum wavelength `lambda_(m)` at temperature `2000 K`. Its corresponding wavelength at temperature 3000 will be

Options

A. `3/2 lambda_(m)`
B. `2/3 lambda_(m)`
C. `16/81lambda_(m)`
D. `81/16 lambda_(m)`

Explanation

To find the maximum wavelength \(\lambda_m'\) at a temperature of \(3000 \, K\) given that the maximum wavelength \(\lambda_m\) at \(2000 \, K\) is known, we can use Wien's Displacement Law. Here are the steps to solve the problem:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement Law states that the maximum wavelength \(\lambda_m\) of radiation emitted by a black body is inversely proportional to its absolute temperature \(T\). Mathematically, it can be expressed as:
\[ \lambda_m = \frac{b}{T} \]
where \(b\) is Wien's displacement constant.

### Step 2: Write the equations for both temperatures
For the first case at \(2000 \, K\):
\[ \lambda_m = \frac{b}{2000} \]

For the second case at \(3000 \, K\):
\[ \lambda_m' = \frac{b}{3000} \]

### Step 3: Set up the ratio of the two wavelengths
To find the relationship between \(\lambda_m\) and \(\lambda_m'\), we can divide the two equations:
\[ \frac{\lambda_m}{\lambda_m'} = \frac{b/2000}{b/3000} \]

### Step 4: Simplify the ratio
The \(b\) cancels out, so we have:
\[ \frac{\lambda_m}{\lambda_m'} = \frac{3000}{2000} = \frac{3}{2} \]

### Step 5: Solve for \(\lambda_m'\)
Rearranging this gives:
\[ \lambda_m' = \frac{2}{3} \lambda_m \]

### Conclusion
Thus, the maximum wavelength at \(3000 \, K\) is:
\[ \lambda_m' = \frac{2}{3} \lambda_m \]

### Final Answer
If \(\lambda_m\) is the maximum wavelength at \(2000 \, K\), then at \(3000 \, K\), the maximum wavelength \(\lambda_m'\) will be \(\frac{2}{3} \lambda_m\).

---