the maximmum temperature reached during an atomic explosion was of the order of `10^(7)` K. Calculate the wavelength of maximum energy b = 0.293 cm K.

Explanation

To solve the problem of finding the wavelength of maximum energy during an atomic explosion at a temperature of \(10^7\) K, we will use Wien's Displacement Law. Here’s a step-by-step solution:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement Law states that the wavelength of maximum energy (\(\lambda_{max}\)) is inversely proportional to the temperature (T) of the black body. Mathematically, it is expressed as:
\[ \lambda_{max} \cdot T = b \]
where \(b\) is Wien's displacement constant.

### Step 2: Identify the Given Values
From the problem, we have:
- Maximum temperature, \(T = 10^7 \, \text{K}\)
- Wien's displacement constant, \(b = 0.293 \, \text{cm \cdot K}\)

### Step 3: Convert the Constant to Meters
Since we want the wavelength in meters, we need to convert \(b\) from centimeters to meters:
\[ b = 0.293 \, \text{cm} = 0.293 \times 10^{-2} \, \text{m} = 2.93 \times 10^{-3} \, \text{m \cdot K} \]

### Step 4: Apply Wien's Displacement Law
Now, we can rearrange the formula to find \(\lambda_{max}\):
\[ \lambda_{max} = \frac{b}{T} \]

### Step 5: Substitute the Values
Substituting the values of \(b\) and \(T\) into the equation:
\[ \lambda_{max} = \frac{2.93 \times 10^{-3} \, \text{m \cdot K}}{10^7 \, \text{K}} \]

### Step 6: Calculate \(\lambda_{max}\)
Perform the calculation:
\[ \lambda_{max} = \frac{2.93 \times 10^{-3}}{10^7} = 2.93 \times 10^{-10} \, \text{m} \]

### Step 7: Convert to Angstroms
Since \(1 \, \text{Angstrom} = 10^{-10} \, \text{m}\), we can convert the wavelength to Angstroms:
\[ \lambda_{max} = 2.93 \, \text{Angstroms} \]

### Final Answer
The wavelength of maximum energy is approximately:
\[ \lambda_{max} \approx 2.93 \, \text{Angstroms} \]

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