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. (a)3 nC
B. (b)`1/3 nC`
C. (c)2 nC
D. (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 the Given Values**:
- Radius of the metal ball, \( r = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \)
- Dielectric strength of air, \( E = 3 \times 10^{6} \text{ V/m} \)

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

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

4. **Substituting the Values**:
Now substitute the known values into the equation:
\[ q = (3 \times 10^{6}) \cdot (4 \pi \cdot 8.85 \times 10^{-12}) \cdot (1 \times 10^{-3})^2 \]

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

6. **Calculating \( 4 \pi \epsilon_0 \)**:
Calculate \( 4 \pi \epsilon_0 \):
\[ 4 \pi \cdot 8.85 \times 10^{-12} \approx 1.112 \times 10^{-10} \text{ C}^2/\text{N m}^2 \]

7. **Final Calculation**:
Now plug everything back into the equation:
\[ q = (3 \times 10^{6}) \cdot (1.112 \times 10^{-10}) \cdot (1 \times 10^{-6}) \]
\[ q = 3.336 \times 10^{-10} \text{ C} \]

8. **Converting to NanoCoulombs**:
Convert the charge to nanoCoulombs:
\[ q \approx 0.3336 \text{ nC} \approx 0.33 \text{ nC} \]

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

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

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