The maximum radiant energyy emitted at 1000 K is for a wavelength of 2.9 Å, the maximum radiant energy emitted at 2000 K will be for a wavelength of

Options

A. 29000 Å
B. 14500 Å
C. 1.45 Å
D. 7250 Å

Explanation

To solve the problem of finding the maximum radiant energy emitted at 2000 K based on the given information at 1000 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 (λ_max) at which the maximum radiant energy is emitted is inversely proportional to the temperature (T) of the black body. Mathematically, it can be expressed as:
\[ \lambda_{max} \cdot T = b \]
where \( b \) is a constant.

### Step 2: Write the relationship for two different temperatures
For two different temperatures, we can write:
\[ \lambda_1 \cdot T_1 = \lambda_2 \cdot T_2 \]
where:
- \( \lambda_1 \) is the wavelength at temperature \( T_1 \)
- \( \lambda_2 \) is the wavelength at temperature \( T_2 \)

### Step 3: Substitute the known values
From the problem, we know:
- \( \lambda_1 = 2.9 \) Å (wavelength at 1000 K)
- \( T_1 = 1000 \) K
- \( T_2 = 2000 \) K

Now we can substitute these values into the equation:
\[ 2.9 \, \text{Å} \cdot 1000 \, \text{K} = \lambda_2 \cdot 2000 \, \text{K} \]

### Step 4: Solve for \( \lambda_2 \)
Rearranging the equation to find \( \lambda_2 \):
\[ \lambda_2 = \frac{2.9 \, \text{Å} \cdot 1000 \, \text{K}}{2000 \, \text{K}} \]

### Step 5: Calculate \( \lambda_2 \)
Now, we can calculate \( \lambda_2 \):
\[ \lambda_2 = \frac{2.9 \cdot 1000}{2000} \]
\[ \lambda_2 = \frac{2900}{2000} \]
\[ \lambda_2 = 1.45 \, \text{Å} \]

### Conclusion
The maximum radiant energy emitted at 2000 K will be for a wavelength of **1.45 Å**.

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