Two bulba A and B of 200 W and 25 W, respectively are designed for the same voltage wire used in both the cases are same. The ratio of radius of cross section of filament of bulb A to that of the bulb B is

Options

A. `2 sqrt(3)`
B. `2 sqrt(2)`
C. 2
D. 4

Explanation

To solve the problem of finding the ratio of the radius of the cross-section of the filaments of two bulbs A and B, we can follow these steps:

### Step 1: Understand the relationship between power, voltage, and resistance.
The power \( P \) of an electrical device is given by the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage and \( R \) is the resistance.

### Step 2: Write the equations for both bulbs.
For bulb A (200 W):
\[ P_A = \frac{V^2}{R_A} \implies R_A = \frac{V^2}{P_A} \]
For bulb B (25 W):
\[ P_B = \frac{V^2}{R_B} \implies R_B = \frac{V^2}{P_B} \]

### Step 3: Find the ratio of the resistances.
Taking the ratio of the resistances:
\[ \frac{R_A}{R_B} = \frac{P_B}{P_A} = \frac{25}{200} = \frac{1}{8} \]

### Step 4: Relate resistance to the physical dimensions of the filament.
The resistance \( R \) of a cylindrical conductor is given by:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area. For a circular cross-section, the area \( A \) is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius.

### Step 5: Substitute the area into the resistance formula.
Thus, we can express the resistances of the bulbs in terms of their radii:
\[ R_A = \frac{\rho L}{\pi r_A^2} \quad \text{and} \quad R_B = \frac{\rho L}{\pi r_B^2} \]

### Step 6: Find the ratio of the resistances in terms of the radii.
Substituting these into the ratio we found earlier:
\[ \frac{R_A}{R_B} = \frac{\frac{\rho L}{\pi r_A^2}}{\frac{\rho L}{\pi r_B^2}} = \frac{r_B^2}{r_A^2} \]
Since we know \( \frac{R_A}{R_B} = \frac{1}{8} \):
\[ \frac{r_B^2}{r_A^2} = \frac{1}{8} \]

### Step 7: Solve for the ratio of the radii.
Taking the square root of both sides gives:
\[ \frac{r_A}{r_B} = \sqrt{8} = 2\sqrt{2} \]

### Final Answer:
The ratio of the radius of the cross-section of filament of bulb A to that of bulb B is:
\[ \frac{r_A}{r_B} = 2\sqrt{2} \]