If at temperature `T_(1)=1000K`, the wavelength is `2.2xx10^(-4)` m, then at what temperature the wavelength will be `4.4xx10^(-5)m`?

Options

A. 5200 K
B. 4700 K
C. 5000 K
D. 4800 K

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the product of the temperature (T) of a black body and the wavelength (λ) of its peak emission is a constant. This can be expressed as:

\[ T \cdot \lambda = B \]

where \( B \) is a constant.

Given:
- \( T_1 = 1000 \, K \)
- \( \lambda_1 = 2.2 \times 10^{-4} \, m \)
- \( \lambda_2 = 4.4 \times 10^{-5} \, m \)

We need to find \( T_2 \).

### Step 1: Write the relationship using Wien's Displacement Law
From Wien's Displacement Law, we know:

\[ T_1 \cdot \lambda_1 = T_2 \cdot \lambda_2 \]

### Step 2: Rearrange the equation to solve for \( T_2 \)
Rearranging the equation gives us:

\[ T_2 = \frac{T_1 \cdot \lambda_1}{\lambda_2} \]

### Step 3: Substitute the known values into the equation
Now we can substitute the known values into the equation:

\[ T_2 = \frac{1000 \, K \cdot (2.2 \times 10^{-4} \, m)}{4.4 \times 10^{-5} \, m} \]

### Step 4: Simplify the equation
Calculating the right-hand side:

1. Calculate the fraction of the wavelengths:

\[ \frac{2.2 \times 10^{-4}}{4.4 \times 10^{-5}} = \frac{2.2}{4.4} \times \frac{10^{-4}}{10^{-5}} = 0.5 \times 10^{1} = 5 \]

2. Now substitute this back into the equation for \( T_2 \):

\[ T_2 = 1000 \, K \cdot 5 = 5000 \, K \]

### Final Answer
Thus, the temperature \( T_2 \) at which the wavelength will be \( 4.4 \times 10^{-5} \, m \) is:

\[ T_2 = 5000 \, K \]

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