Estimate the temperature at which a body may appear blue or red. The values of `lambda_(mean)` for these are `5000` and `7500Å` respectively. [Given Wein's constant `b = 0.3 cmK]`

Explanation

To estimate the temperature at which a body may appear blue or red using Wien's Law, we can follow these steps:

### Step 1: Understand Wien's Law
Wien's Law states that the wavelength at which the emission of a black body spectrum is maximized is inversely proportional to the temperature of the body. Mathematically, it can be expressed as:
\[ \lambda_{mean} \cdot T = b \]
where:
- \( \lambda_{mean} \) is the wavelength in centimeters,
- \( T \) is the temperature in Kelvin,
- \( b \) is Wien's constant (given as \( 0.3 \, \text{cm} \cdot \text{K} \)).

### Step 2: Convert Wavelengths to Centimeters
The given wavelengths are in Angstroms (Å). We need to convert them to centimeters:
- \( 1 \, \text{Å} = 10^{-8} \, \text{cm} \)
- For blue light:
\[ \lambda_{blue} = 5000 \, \text{Å} = 5000 \times 10^{-8} \, \text{cm} = 5 \times 10^{-5} \, \text{cm} \]
- For red light:
\[ \lambda_{red} = 7500 \, \text{Å} = 7500 \times 10^{-8} \, \text{cm} = 7.5 \times 10^{-5} \, \text{cm} \]

### Step 3: Calculate Temperature for Blue Light
Using Wien's Law for blue light:
\[ T_{blue} = \frac{b}{\lambda_{blue}} \]
Substituting the values:
\[ T_{blue} = \frac{0.3 \, \text{cm} \cdot \text{K}}{5 \times 10^{-5} \, \text{cm}} \]
\[ T_{blue} = \frac{0.3}{5 \times 10^{-5}} \]
\[ T_{blue} = 6 \times 10^{3} \, \text{K} \]

### Step 4: Calculate Temperature for Red Light
Using Wien's Law for red light:
\[ T_{red} = \frac{b}{\lambda_{red}} \]
Substituting the values:
\[ T_{red} = \frac{0.3 \, \text{cm} \cdot \text{K}}{7.5 \times 10^{-5} \, \text{cm}} \]
\[ T_{red} = \frac{0.3}{7.5 \times 10^{-5}} \]
\[ T_{red} = 4 \times 10^{3} \, \text{K} \]

### Step 5: Final Results
The estimated temperatures at which a body may appear blue or red are:
- \( T_{blue} = 6000 \, \text{K} \)
- \( T_{red} = 4000 \, \text{K} \)