Two stars radiate maximum energy at wavelengths `3.6xx10^(-5)cm` and `4.8xx10^(-5)cm` respectively. The ratio of their temperature is

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(3)/(5)`
D. `(4)/(3)`

Answer

To find the ratio of temperatures of two stars that radiate maximum energy at given wavelengths, we can use Wien's Displacement Law. The law states that the wavelength at which the emission of a black body spectrum is maximized (λ_max) is inversely proportional to the temperature (T) of the black body. The relationship can be expressed as:

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

where \( b \) is Wien's displacement constant.

### Step-by-Step Solution:

1. **Identify the given wavelengths**:
- For the first star, \( \lambda_1 = 3.6 \times 10^{-5} \) cm
- For the second star, \( \lambda_2 = 4.8 \times 10^{-5} \) cm

2. **Convert the wavelengths to meters** (if necessary):
- \( \lambda_1 = 3.6 \times 10^{-5} \) cm = \( 3.6 \times 10^{-7} \) m
- \( \lambda_2 = 4.8 \times 10^{-5} \) cm = \( 4.8 \times 10^{-7} \) m

3. **Apply Wien's Law**:
According to Wien's Law:
\[ \lambda_{1} \cdot T_1 = b \quad \text{and} \quad \lambda_{2} \cdot T_2 = b \]

4. **Set up the ratio of temperatures**:
Since both expressions equal \( b \), we can set them equal to each other:
\[ \lambda_1 \cdot T_1 = \lambda_2 \cdot T_2 \]

Rearranging gives:
\[ \frac{T_1}{T_2} = \frac{\lambda_2}{\lambda_1} \]

5. **Substitute the values of wavelengths**:
\[ \frac{T_1}{T_2} = \frac{4.8 \times 10^{-7}}{3.6 \times 10^{-7}} \]

6. **Simplify the ratio**:
\[ \frac{T_1}{T_2} = \frac{4.8}{3.6} = \frac{48}{36} = \frac{4}{3} \]

7. **Conclusion**:
Therefore, the ratio of the temperatures of the two stars is:
\[ T_1 : T_2 = 4 : 3 \]

### Final Answer:
The ratio of their temperatures is \( 4 : 3 \).