Two stars A and B radiate maximum energy 5200Å and 6500Å respectively. Then the ratio of absolute temperatures of A and B is

Options

A. `25 : 16`
B. `5 : 4`
C. `4 : 5`
D. `16 : 25`

Explanation

To solve the problem of finding the ratio of absolute temperatures of two stars A and B, we can use Wien's Displacement Law, which states that the product of the maximum wavelength (\( \lambda_{max} \)) and the absolute temperature (T) of a black body is a constant. The formula can be expressed as:

\[ \lambda_{max} \cdot T = b \]

where \( b \) is a constant approximately equal to \( 3 \times 10^{-3} \) m·K.

Given:
- For star A, \( \lambda_{max1} = 5200 \) Å
- For star B, \( \lambda_{max2} = 6500 \) Å

### Step 1: Convert wavelengths from Ångströms to meters
1 Å = \( 10^{-10} \) m, so:
- \( \lambda_{max1} = 5200 \times 10^{-10} \) m
- \( \lambda_{max2} = 6500 \times 10^{-10} \) m

### Step 2: Write the equations according to Wien's Law
For star A:
\[ \lambda_{max1} \cdot T_A = b \]
For star B:
\[ \lambda_{max2} \cdot T_B = b \]

### Step 3: Set the equations equal to each other
Since both equations equal \( b \), we can set them equal to each other:
\[ \lambda_{max1} \cdot T_A = \lambda_{max2} \cdot T_B \]

### Step 4: Rearrange to find the ratio of temperatures
Rearranging gives:
\[ \frac{T_A}{T_B} = \frac{\lambda_{max2}}{\lambda_{max1}} \]

### Step 5: Substitute the values of wavelengths
Substituting the values we have:
\[ \frac{T_A}{T_B} = \frac{6500 \times 10^{-10}}{5200 \times 10^{-10}} = \frac{6500}{5200} \]

### Step 6: Simplify the ratio
Now simplify \( \frac{6500}{5200} \):
\[ \frac{6500}{5200} = \frac{65}{52} = \frac{13}{10.4} = \frac{5}{4} \]

### Final Result
Thus, the ratio of absolute temperatures of stars A and B is:
\[ \frac{T_A}{T_B} = \frac{5}{4} \]