The maximum energy in thermal radiation from a source occurs at the wavelength 4000Å. The effective temperature of the source

Options

A. 7000 K
B. 80000 K
C. `10^(4)` K
D. `10^(6)` K

Explanation

To find the effective temperature of the source based on the maximum energy in thermal radiation occurring at a wavelength of 4000 Å, we can use Wien's displacement law. Here’s the step-by-step solution:

### Step 1: Understand Wien's Law
Wien's Law states that the wavelength at which the intensity of radiation is maximum (λm) is inversely proportional to the absolute temperature (T) of the black body. The formula is given by:

\[ T = \frac{b}{\lambda_m} \]

where:
- \( T \) is the absolute temperature in Kelvin,
- \( b \) is Wien's displacement constant, approximately \( 2.89 \times 10^{-3} \, \text{m} \cdot \text{K} \),
- \( \lambda_m \) is the wavelength in meters.

### Step 2: Convert Wavelength from Angstroms to Meters
The given wavelength is 4000 Å. To convert this to meters:

\[ \lambda_m = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4.0 \times 10^{-7} \, \text{m} \]

### Step 3: Substitute Values into the Formula
Now, substitute the values of \( b \) and \( \lambda_m \) into the formula:

\[ T = \frac{2.89 \times 10^{-3}}{4.0 \times 10^{-7}} \]

### Step 4: Calculate the Temperature
Now, perform the calculation:

\[ T = \frac{2.89 \times 10^{-3}}{4.0 \times 10^{-7}} = 7225 \, \text{K} \]

### Step 5: Round the Temperature
Since the question asks for the effective temperature, we can round this to the nearest thousand:

\[ T \approx 7000 \, \text{K} \]

### Conclusion
Thus, the effective temperature of the source is approximately 7000 Kelvin.

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