Two stars radiate maximum energy at wavelength `3.6xx10^(-7)` m and `4.8xx10^(-7)` m respectively. What is the ratio of their temperatures ?

Explanation

To find the ratio of the temperatures of two stars that radiate maximum energy at different wavelengths, we can use Wien's Displacement Law. Here’s a step-by-step solution:

### Step 1: Identify the wavelengths
Let:
- \( \lambda_{m1} = 3.6 \times 10^{-7} \, \text{m} \) (wavelength for star 1)
- \( \lambda_{m2} = 4.8 \times 10^{-7} \, \text{m} \) (wavelength for star 2)

### Step 2: Apply Wien's Displacement Law
Wien's Displacement Law states that the wavelength corresponding to maximum energy (\( \lambda_m \)) is inversely proportional to the temperature (T) of the black body. Mathematically, this can be expressed as:
\[ \lambda_m \cdot T = b \]
where \( b \) is a constant.

From this, we can write:
\[ \lambda_{m1} \cdot T_1 = \lambda_{m2} \cdot T_2 \]

### Step 3: Rearrange the equation to find the temperature ratio
We need to find the ratio of the temperatures \( \frac{T_1}{T_2} \):
\[ \frac{T_1}{T_2} = \frac{\lambda_{m2}}{\lambda_{m1}} \]

### Step 4: Substitute the values of the wavelengths
Substituting the values of \( \lambda_{m1} \) and \( \lambda_{m2} \):
\[ \frac{T_1}{T_2} = \frac{4.8 \times 10^{-7}}{3.6 \times 10^{-7}} \]

### Step 5: Simplify the ratio
The \( 10^{-7} \) in the numerator and denominator cancels out:
\[ \frac{T_1}{T_2} = \frac{4.8}{3.6} \]

Now, simplify \( \frac{4.8}{3.6} \):
- Dividing both the numerator and denominator by 1.2:
\[ \frac{4.8 \div 1.2}{3.6 \div 1.2} = \frac{4}{3} \]

### Final Answer
Thus, the ratio of the temperatures \( \frac{T_1}{T_2} \) is:
\[ T_1 : T_2 = 4 : 3 \]

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