Two stars emit maximum radiation at wavelength 3600 Å and 4800 Å respectively. The ratio of their temperatures is

Options

A. `1 : 2`
B. `3 : 4`
C. `4 : 3`
D. `2 : 1`

Answer

To find the ratio of the temperatures of two stars that emit maximum radiation at different wavelengths, we can 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 at which the emission of a black body spectrum is maximized (λ_max) is inversely proportional to the absolute temperature (T) of the body. Mathematically, it can be expressed as:
\[ \lambda_{max} = \frac{K}{T} \]
where K is Wien's constant.

### Step 2: Set Up the Equations
For the two stars, we can write:
- For Star 1 (λ_m1 = 3600 Å):
\[ \lambda_{m1} = \frac{K}{T_1} \]
- For Star 2 (λ_m2 = 4800 Å):
\[ \lambda_{m2} = \frac{K}{T_2} \]

### Step 3: Form the Ratio of Wavelengths
We can form the ratio of the two equations:
\[ \frac{\lambda_{m1}}{\lambda_{m2}} = \frac{K/T_1}{K/T_2} \]
The K cancels out, leading to:
\[ \frac{\lambda_{m1}}{\lambda_{m2}} = \frac{T_2}{T_1} \]

### Step 4: Substitute the Given Wavelengths
Now, substitute the values of λ_m1 and λ_m2:
\[ \frac{3600 \, \text{Å}}{4800 \, \text{Å}} = \frac{T_2}{T_1} \]

### Step 5: Simplify the Ratio
Now simplify the left side:
\[ \frac{3600}{4800} = \frac{3}{4} \]
Thus, we have:
\[ \frac{3}{4} = \frac{T_2}{T_1} \]

### Step 6: Find the Ratio of Temperatures
To find the ratio of temperatures \( T_1 \) to \( T_2 \), we can take the reciprocal:
\[ \frac{T_1}{T_2} = \frac{4}{3} \]

### Final Answer
The ratio of the temperatures of the two stars is:
\[ T_1 : T_2 = 4 : 3 \]

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