The insulation property of air breaks down at `E=3xx10^(6) "volt"//meter`. The maximum charge that can be given to a sphere of diameter `5m` is approximately (in coulombs)

Options

A. `2xx10^(-2)`
B. `2xx10^(-3)`
C. `2xx10^(-4)`
D. `2xx10^(-5)`

Explanation

To find the maximum charge that can be given to a sphere of diameter 5 meters before the insulation property of air breaks down, we can follow these steps:

### Step 1: Understand the Given Information
- The breakdown electric field of air is given as \( E = 3 \times 10^6 \, \text{V/m} \).
- The diameter of the sphere is \( 5 \, \text{m} \), which means the radius \( r \) is \( \frac{5}{2} = 2.5 \, \text{m} \).

### Step 2: Use the Formula for Electric Field Due to a Sphere
The electric field \( E \) at the surface of a charged sphere is given by the formula:
\[ E = \frac{k \cdot Q}{r^2} \]
where:
- \( E \) is the electric field,
- \( k \) is Coulomb's constant (\( k \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( Q \) is the charge on the sphere,
- \( r \) is the radius of the sphere.

### Step 3: Rearrange the Formula to Solve for Charge \( Q \)
Rearranging the formula gives:
\[ Q = \frac{E \cdot r^2}{k} \]

### Step 4: Substitute the Values
Now, we can substitute the known values into the equation:
- \( E = 3 \times 10^6 \, \text{V/m} \)
- \( r = 2.5 \, \text{m} \)
- \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)

Substituting these values:
\[ Q = \frac{(3 \times 10^6) \cdot (2.5)^2}{9 \times 10^9} \]

### Step 5: Calculate \( r^2 \)
Calculating \( r^2 \):
\[ (2.5)^2 = 6.25 \]

### Step 6: Substitute \( r^2 \) Back into the Equation
Now substituting \( r^2 \) back into the equation:
\[ Q = \frac{(3 \times 10^6) \cdot 6.25}{9 \times 10^9} \]

### Step 7: Calculate the Numerator
Calculating the numerator:
\[ 3 \times 10^6 \cdot 6.25 = 18.75 \times 10^6 \]

### Step 8: Final Calculation
Now, substituting this into the equation for \( Q \):
\[ Q = \frac{18.75 \times 10^6}{9 \times 10^9} = 2.08 \times 10^{-3} \, \text{C} \]

### Conclusion
The maximum charge that can be given to the sphere is approximately:
\[ Q \approx 2.08 \times 10^{-3} \, \text{C} \]