The temperature of an elactric bulb changes from 2000K to 3000K due to a.c. voltage fluctuations. Calculate the percentage rise in electric power consumed ?

Explanation

To calculate the percentage rise in electric power consumed when the temperature of an electric bulb changes from 2000 K to 3000 K, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between power and temperature**:
According to Stefan-Boltzmann Law, the power radiated by a black body is proportional to the fourth power of its absolute temperature:
\[ P = k \cdot T^4 \]
where \( P \) is the power, \( k \) is a constant (which includes the Stefan-Boltzmann constant and the area), and \( T \) is the temperature in Kelvin.

2. **Calculate initial and final power**:
- For the initial power \( P_1 \) at \( T_1 = 2000 \, K \):
\[ P_1 = k \cdot (2000)^4 \]
- For the final power \( P_2 \) at \( T_2 = 3000 \, K \):
\[ P_2 = k \cdot (3000)^4 \]

3. **Find the change in power**:
The change in power \( \Delta P \) is given by:
\[ \Delta P = P_2 - P_1 = k \cdot (3000^4) - k \cdot (2000^4) \]
Factoring out \( k \):
\[ \Delta P = k \cdot \left(3000^4 - 2000^4\right) \]

4. **Calculate the percentage rise in power**:
The percentage rise in power is given by:
\[ \text{Percentage Rise} = \frac{\Delta P}{P_1} \times 100 \]
Substituting the expressions for \( \Delta P \) and \( P_1 \):
\[ \text{Percentage Rise} = \frac{k \cdot (3000^4 - 2000^4)}{k \cdot (2000^4)} \times 100 \]
The \( k \) cancels out:
\[ \text{Percentage Rise} = \frac{3000^4 - 2000^4}{2000^4} \times 100 \]

5. **Calculate \( 3000^4 \) and \( 2000^4 \)**:
- \( 3000^4 = (3 \times 10^3)^4 = 81 \times 10^{12} \)
- \( 2000^4 = (2 \times 10^3)^4 = 16 \times 10^{12} \)

6. **Substitute these values back into the percentage rise formula**:
\[ \text{Percentage Rise} = \frac{81 \times 10^{12} - 16 \times 10^{12}}{16 \times 10^{12}} \times 100 \]
\[ = \frac{65 \times 10^{12}}{16 \times 10^{12}} \times 100 \]
\[ = \frac{65}{16} \times 100 \]
\[ = 4.0625 \times 100 = 406.25\% \]

### Final Answer:
The percentage rise in electric power consumed is approximately **406.25%**.