A hollow metal sphere is charged with `0.4muC` of charge and has a radius of 0.1 find the potential (i) at the surface (ii) inside the sphere (iii) at a distance of 0.6m from the centre. The sphere is placed in air.

Answer

To solve the problem of finding the electric potential at various points around a hollow metal sphere charged with \(0.4 \, \mu C\) and having a radius of \(0.1 \, m\), we can follow these steps:

### Given:
- Charge, \(Q = 0.4 \, \mu C = 0.4 \times 10^{-6} \, C\)
- Radius of the sphere, \(R = 0.1 \, m\)
- Coulomb's constant, \(k = 9 \times 10^9 \, N \cdot m^2/C^2\)

### (i) Potential at the surface of the sphere

The potential \(V\) at the surface of a charged sphere is given by the formula:

\[ V = \frac{kQ}{R} \]

Substituting the values:

\[ V = \frac{9 \times 10^9 \times 0.4 \times 10^{-6}}{0.1} \]

Calculating the numerator:

\[ 9 \times 10^9 \times 0.4 \times 10^{-6} = 3.6 \times 10^4 \]

Now, dividing by \(0.1\):

\[ V = \frac{3.6 \times 10^4}{0.1} = 3.6 \times 10^5 \, V = 36000 \, V \]

### (ii) Potential inside the sphere

For a hollow metal sphere, the potential inside the sphere is the same as the potential at the surface. Therefore:

\[ V_{\text{inside}} = V_{\text{surface}} = 36000 \, V \]

### (iii) Potential at a distance of \(0.6 \, m\) from the center

For points outside the sphere (i.e., at a distance greater than the radius of the sphere), the potential is given by the same formula:

\[ V = \frac{kQ}{r} \]

where \(r\) is the distance from the center of the sphere. Here, \(r = 0.6 \, m\).

Substituting the values:

\[ V = \frac{9 \times 10^9 \times 0.4 \times 10^{-6}}{0.6} \]

Calculating the numerator (as before):

\[ 9 \times 10^9 \times 0.4 \times 10^{-6} = 3.6 \times 10^4 \]

Now, dividing by \(0.6\):

\[ V = \frac{3.6 \times 10^4}{0.6} = 6 \times 10^4 \, V = 6000 \, V \]

### Summary of Results:
- (i) Potential at the surface: \(36000 \, V\)
- (ii) Potential inside the sphere: \(36000 \, V\)
- (iii) Potential at a distance of \(0.6 \, m\) from the center: \(6000 \, V\)