A hollow charged metal sphere has radius `r`. If the potential difference between its surface and a point at a distance `3r` from the centre is V, then electric field intensity at a distance `3r` is

Options

A. `(V)/(2r)`
B. `(V)/(3r)`
C. `(V)/(6r)`
D. `(V)/(4r)`

Answer

To solve the problem, we need to find the electric field intensity at a distance of `3r` from the center of a hollow charged metal sphere, given that the potential difference between its surface and this point is `V`.

### Step-by-Step Solution:

1. **Understanding the Potential of a Hollow Charged Sphere**:
- For a hollow charged metal sphere, the electric potential \( V \) at any point outside the sphere (at a distance \( r \) from the center) is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge of the sphere, and \( r \) is the distance from the center of the sphere.

2. **Potential at the Surface of the Sphere**:
- The radius of the sphere is given as \( r \). Therefore, the potential at the surface of the sphere (at distance \( r \)) is:
\[ V_{\text{surface}} = \frac{kQ}{r} \]

3. **Potential at a Distance of \( 3r \)**:
- Now, we need to find the potential at a distance of \( 3r \) from the center of the sphere:
\[ V_{3r} = \frac{kQ}{3r} \]

4. **Calculating the Potential Difference**:
- The potential difference \( V \) between the surface of the sphere and the point at a distance of \( 3r \) is given by:
\[ V = V_{\text{surface}} - V_{3r} \]
Substituting the values we found:
\[ V = \frac{kQ}{r} - \frac{kQ}{3r} \]
- To simplify this, we can factor out \( \frac{kQ}{r} \):
\[ V = \frac{kQ}{r} \left(1 - \frac{1}{3}\right) = \frac{kQ}{r} \cdot \frac{2}{3} = \frac{2kQ}{3r} \]

5. **Finding the Electric Field Intensity**:
- The electric field intensity \( E \) at a distance \( r \) from a point charge is given by:
\[ E = -\frac{dV}{dr} \]
- For a hollow charged sphere, outside the sphere, the electric field can also be expressed as:
\[ E = \frac{kQ}{r^2} \]
- Therefore, at a distance of \( 3r \):
\[ E_{3r} = \frac{kQ}{(3r)^2} = \frac{kQ}{9r^2} \]

### Final Answer:
The electric field intensity at a distance of \( 3r \) from the center of the hollow charged metal sphere is:
\[ E_{3r} = \frac{kQ}{9r^2} \]