If temperature of ideal black body is decreased from T to T/2 than find out percentage loss in emissive rate

Explanation

To solve the problem of finding the percentage loss in the emissive rate of an ideal black body when its temperature is decreased from \( T \) to \( \frac{T}{2} \), we can follow these steps:

### Step 1: Understand the Emissive Rate
The emissive rate \( E \) of a black body is proportional to the fourth power of its absolute temperature \( T \). Mathematically, we can express this as:
\[ E \propto T^4 \]

### Step 2: Calculate the Emissive Rate at Initial Temperature
Let the emissive rate at the initial temperature \( T \) be \( E_1 \):
\[ E_1 = k \cdot T^4 \]
where \( k \) is a proportionality constant.

### Step 3: Calculate the Emissive Rate at Reduced Temperature
When the temperature is decreased to \( \frac{T}{2} \), the new emissive rate \( E_2 \) can be calculated as:
\[ E_2 = k \cdot \left(\frac{T}{2}\right)^4 \]
Calculating \( E_2 \):
\[ E_2 = k \cdot \frac{T^4}{16} \]

### Step 4: Find the Loss in Emissive Rate
The loss in emissive rate can be calculated as:
\[ \text{Loss} = E_1 - E_2 \]
Substituting the values:
\[ \text{Loss} = k \cdot T^4 - k \cdot \frac{T^4}{16} \]
Factoring out \( k \cdot T^4 \):
\[ \text{Loss} = k \cdot T^4 \left(1 - \frac{1}{16}\right) = k \cdot T^4 \cdot \frac{15}{16} \]

### Step 5: Calculate the Percentage Loss
The percentage loss in emissive rate is given by:
\[ \text{Percentage Loss} = \frac{\text{Loss}}{E_1} \times 100\% \]
Substituting the values:
\[ \text{Percentage Loss} = \frac{k \cdot T^4 \cdot \frac{15}{16}}{k \cdot T^4} \times 100\% \]
The \( k \cdot T^4 \) terms cancel out:
\[ \text{Percentage Loss} = \frac{15}{16} \times 100\% \]
Calculating this gives:
\[ \text{Percentage Loss} = 93.75\% \]

### Step 6: Final Result
Thus, the percentage loss in the emissive rate when the temperature is decreased from \( T \) to \( \frac{T}{2} \) is approximately:
\[ \text{Percentage Loss} \approx 93.75\% \]