A spark passes in air when the potential gradient at the surface of charged conductor is `4xx10^(6) Vm^(-1)`. What must be the radius of an insulated metal sphere which can be charged to a potential of `4xx10^(6) V` before sparking into air ?

Explanation

To solve the problem, we need to find the radius of an insulated metal sphere that can be charged to a potential of \(4 \times 10^6 \, V\) before sparking occurs in air, given that the potential gradient at the surface of a charged conductor is \(4 \times 10^6 \, V/m\).

### Step-by-Step Solution:

1. **Understand the Relationship Between Potential and Potential Gradient:**
The potential \(V\) at the surface of a charged conductor is related to the potential gradient \(E\) (which is the electric field) and the radius \(r\) of the sphere by the formula:
\[ E = \frac{V}{r} \]
where \(E\) is the potential gradient (in \(V/m\)), \(V\) is the potential (in \(V\)), and \(r\) is the radius (in \(m\)).

2. **Substitute the Given Values:**
We know from the problem statement that:
- \(E = 4 \times 10^6 \, V/m\)
- \(V = 4 \times 10^6 \, V\)

Substitute these values into the equation:
\[ 4 \times 10^6 = \frac{4 \times 10^6}{r} \]

3. **Rearranging the Equation:**
To find the radius \(r\), we can rearrange the equation:
\[ r = \frac{V}{E} \]

4. **Calculate the Radius:**
Substitute the values of \(V\) and \(E\) into the rearranged equation:
\[ r = \frac{4 \times 10^6}{4 \times 10^6} = 1 \, m \]

5. **Conclusion:**
The radius of the insulated metal sphere that can be charged to a potential of \(4 \times 10^6 \, V\) before sparking occurs is:
\[ r = 1 \, m \]

### Final Answer:
The radius of the insulated metal sphere is \(1 \, m\).