What is the largest charge a metal ball of 1 mm radius can hold ? (Dielectric strength of air is `3xx10^(6) V m^(-1)`)

Options

A. 3 nC
B. `1/3 nC`
C. 2 nC
D. `1/2 nC`

Answer

To find the largest charge a metal ball of 1 mm radius can hold, we will use the concept of dielectric strength and the formula for the electric field around a charged sphere.

### Step-by-Step Solution:

1. **Understand Dielectric Strength**:
The dielectric strength of air is given as \(3 \times 10^6 \, \text{V/m}\). This value represents the maximum electric field that air can withstand before breakdown occurs.

2. **Electric Field Formula**:
The electric field \(E\) around a charged sphere is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{r^2} \]
where:
- \(E\) is the electric field,
- \(Q\) is the charge on the sphere,
- \(r\) is the radius of the sphere,
- \(\epsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{F/m}\).

3. **Rearranging the Formula**:
To find the charge \(Q\), we can rearrange the formula:
\[ Q = E \cdot 4 \pi \epsilon_0 \cdot r^2 \]

4. **Substituting Values**:
Now we will substitute the known values into the equation:
- \(E = 3 \times 10^6 \, \text{V/m}\)
- \(r = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)

Thus, we have:
\[ Q = (3 \times 10^6) \cdot (4 \pi \cdot 8.85 \times 10^{-12}) \cdot (1 \times 10^{-3})^2 \]

5. **Calculating \(r^2\)**:
Calculate \(r^2\):
\[ r^2 = (1 \times 10^{-3})^2 = 1 \times 10^{-6} \, \text{m}^2 \]

6. **Calculating \(4 \pi \epsilon_0\)**:
Now calculate \(4 \pi \epsilon_0\):
\[ 4 \pi \cdot 8.85 \times 10^{-12} \approx 1.11 \times 10^{-10} \, \text{F/m} \]

7. **Final Calculation**:
Now substitute everything back into the equation:
\[ Q = (3 \times 10^6) \cdot (1.11 \times 10^{-10}) \cdot (1 \times 10^{-6}) \]
\[ Q = 3.33 \times 10^{-10} \, \text{C} \]

8. **Convert to NanoCoulombs**:
Since \(1 \, \text{nC} = 10^{-9} \, \text{C}\):
\[ Q = 0.333 \, \text{nC} \approx \frac{1}{3} \, \text{nC} \]

### Conclusion:
The largest charge a metal ball of 1 mm radius can hold is approximately \(0.33 \, \text{nC}\).