A black body emits maximum radiation of wavelength `lambda_(1)` at a certain temperature `T_(1)`. On increasing the temperature, the total energy of radiation emitted is increased 16 times at temperature `T_(2)`. If `lambda_(2)` is the wavelength corresponding to which maximum radiation is emitted at temperature `T_(2)`. Calculate the value of `((lambda_(1))/(lambda_(2)))`.

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Wavelength of maximum radiation at temperature \( T_1 \): \( \lambda_1 \)
- Energy emitted at temperature \( T_1 \): \( E \)
- Energy emitted at temperature \( T_2 \): \( 16E \)
- Wavelength of maximum radiation at temperature \( T_2 \): \( \lambda_2 \)

### Step 2: Use the Energy Relation
According to the Stefan-Boltzmann Law, the energy emitted by a black body is proportional to the fourth power of its absolute temperature:
\[ E \propto T^4 \]
Thus, we can write:
\[ E \propto T_1^4 \quad \text{and} \quad 16E \propto T_2^4 \]
From this, we can express the relationship between \( T_1 \) and \( T_2 \):
\[ \frac{E}{16E} = \frac{T_1^4}{T_2^4} \implies \frac{1}{16} = \left(\frac{T_1}{T_2}\right)^4 \]
Taking the fourth root of both sides gives:
\[ \frac{T_1}{T_2} = \frac{1}{2} \]
This implies:
\[ T_1 = \frac{1}{2} T_2 \quad \text{(Equation 1)} \]

### Step 3: Apply Wien's Displacement Law
Wien's Law states that the product of the wavelength of maximum emission and the absolute temperature is a constant:
\[ \lambda_{\text{max}} T = b \]
For our two temperatures, we can write:
\[ \lambda_1 T_1 = \lambda_2 T_2 \quad \text{(Equation 2)} \]

### Step 4: Substitute \( T_1 \) in Terms of \( T_2 \)
From Equation 1, we can substitute \( T_1 \) in Equation 2:
\[ \lambda_1 \left(\frac{1}{2} T_2\right) = \lambda_2 T_2 \]
Dividing both sides by \( T_2 \) (assuming \( T_2 \neq 0 \)):
\[ \frac{\lambda_1}{2} = \lambda_2 \]

### Step 5: Solve for the Ratio \( \frac{\lambda_1}{\lambda_2} \)
Rearranging the equation gives:
\[ \lambda_1 = 2 \lambda_2 \]
Thus, the ratio \( \frac{\lambda_1}{\lambda_2} \) is:
\[ \frac{\lambda_1}{\lambda_2} = 2 \]

### Final Answer
The value of \( \frac{\lambda_1}{\lambda_2} \) is \( 2 \).

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