Two electric bulbs have filaments of lengths `L` and `2L` diameters `2d` and d and emissivities `3e` and 4e If their temperatures are in the ratio 2:3 their powers will be in the ratio of .

Options

A. `8:27`
B. `4:27`
C. `8:3`
D. `4:9`

Explanation

To find the ratio of powers of the two electric bulbs, we can use the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. The formula for power \( P \) can be expressed as:

\[ P = \varepsilon \sigma A T^4 \]

where:
- \( \varepsilon \) is the emissivity,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the absolute temperature.

### Step 1: Determine the Surface Area of Each Bulb

For a cylindrical filament, the surface area \( A \) can be calculated using the formula:

\[ A = \text{length} \times \text{circumference} = L \times \pi D \]

For Bulb 1:
- Length \( L_1 = L \)
- Diameter \( D_1 = 2d \)
- Circumference \( = \pi D_1 = \pi (2d) = 2\pi d \)
- Surface Area \( A_1 = L \times 2\pi d = 2\pi d L \)

For Bulb 2:
- Length \( L_2 = 2L \)
- Diameter \( D_2 = d \)
- Circumference \( = \pi D_2 = \pi d \)
- Surface Area \( A_2 = 2L \times \pi d = 2\pi d (2L) = 4\pi d L \)

### Step 2: Write the Power Expressions for Each Bulb

Using the surface areas calculated:

For Bulb 1:
\[ P_1 = \varepsilon_1 \sigma A_1 T_1^4 = (3e) \sigma (2\pi d L) T_1^4 \]

For Bulb 2:
\[ P_2 = \varepsilon_2 \sigma A_2 T_2^4 = (4e) \sigma (4\pi d L) T_2^4 \]

### Step 3: Find the Ratio of Powers

Now, we can find the ratio of the powers \( \frac{P_1}{P_2} \):

\[ \frac{P_1}{P_2} = \frac{(3e) \sigma (2\pi d L) T_1^4}{(4e) \sigma (4\pi d L) T_2^4} \]

The constants \( \sigma \), \( \pi \), \( d \), and \( L \) cancel out:

\[ \frac{P_1}{P_2} = \frac{3e \cdot 2}{4e \cdot 4} \cdot \frac{T_1^4}{T_2^4} = \frac{6}{16} \cdot \frac{T_1^4}{T_2^4} = \frac{3}{8} \cdot \frac{T_1^4}{T_2^4} \]

### Step 4: Substitute the Temperature Ratio

Given that the temperatures are in the ratio \( \frac{T_1}{T_2} = \frac{2}{3} \):

\[ \frac{T_1^4}{T_2^4} = \left(\frac{2}{3}\right)^4 = \frac{16}{81} \]

### Step 5: Final Ratio of Powers

Substituting this back into the power ratio:

\[ \frac{P_1}{P_2} = \frac{3}{8} \cdot \frac{16}{81} = \frac{48}{648} = \frac{2}{27} \]

Thus, the final ratio of the powers of the two bulbs is:

\[ \frac{P_1}{P_2} = \frac{2}{27} \]

### Conclusion

The ratio of the powers of the two bulbs is \( \frac{2}{27} \).

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