The temperature of a furnace is `2324^(@)`C and the intensity is maximum in its radiation spectrum nearly at 12000A . If the intensity in the spectrum of a star is maximum nearly at 4800A, calculate the surface temperature of the star.

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength corresponding to the maximum intensity of radiation (λm) is inversely proportional to the absolute temperature (T) of the body. The relationship can be expressed as:

\[ \lambda_m \cdot T = b \]

where \( b \) is a constant.

### Step-by-Step Solution:

1. **Convert the Temperature of the Furnace to Kelvin:**
The temperature of the furnace is given as \( 2324^\circ C \). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Thus,
\[ T_{furnace} = 2324 + 273 = 2597 \, K \]

2. **Identify the Wavelength for the Furnace:**
The maximum intensity wavelength for the furnace is given as \( \lambda_{m,furnace} = 12000 \, \text{Å} \).

3. **Identify the Wavelength for the Star:**
The maximum intensity wavelength for the star is given as \( \lambda_{m,star} = 4800 \, \text{Å} \).

4. **Set Up the Equation Using Wien's Law:**
According to Wien's Law, we can set up the equation:
\[ \lambda_{m,furnace} \cdot T_{furnace} = \lambda_{m,star} \cdot T_{star} \]
Plugging in the known values:
\[ 12000 \, \text{Å} \cdot 2597 \, K = 4800 \, \text{Å} \cdot T_{star} \]

5. **Solve for the Surface Temperature of the Star:**
Rearranging the equation to solve for \( T_{star} \):
\[ T_{star} = \frac{12000 \, \text{Å} \cdot 2597 \, K}{4800 \, \text{Å}} \]

6. **Calculate the Value:**
Performing the calculation:
\[ T_{star} = \frac{31164000 \, \text{Å} \cdot K}{4800 \, \text{Å}} = 6492.5 \, K \]

### Final Answer:
The surface temperature of the star is approximately \( 6492.5 \, K \).

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