Electric Potential at a Distance R/3 from the Centre of a Hollow Metal Sphere

To determine the electric potential at a distance R/3 from the centre of a hollow metal sphere, we need to consider the following aspects:

1. Electric Potential of a Charged Sphere:
The electric potential at a point outside a charged sphere is the same as that of a point charge located at the sphere's centre. This is due to the spherical symmetry of the charge distribution. For a point charge Q, the electric potential (V) at a distance r from the centre is given by the formula:

V = kQ/r

Where k is the electrostatic constant.

2. Electric Field Inside a Hollow Metal Sphere:
Inside a hollow metal sphere, the electric field is zero. This is because the charges on the surface of the metal sphere redistribute themselves in such a way that the electric field inside cancels out. Therefore, the electric potential is constant inside the sphere.

3. Electric Field at the Surface of a Hollow Metal Sphere:
The electric field at the surface of a hollow metal sphere is given by the formula:

E = σ/ε₀

Where σ is the surface charge density and ε₀ is the permittivity of free space.

4. Electric Potential at a Point R/3 from the Centre:
Since the electric field inside the hollow metal sphere is zero, the electric potential is constant throughout the sphere. Therefore, the electric potential at a distance R/3 from the centre of the sphere is the same as the potential at the surface of the sphere.

5. Calculation:
To calculate the electric potential at a distance R/3 from the centre, we can use the formula for the electric field at the surface of a hollow metal sphere:

E = σ/ε₀

Since the charge Q is distributed on the outer surface of the sphere, the surface charge density σ is given by:

σ = Q/(4πR²)

Substituting this into the formula for the electric field, we get:

E = Q/(4πR²ε₀)

Now, using the formula for electric potential V = kQ/r, we can calculate the potential at a distance R/3 from the centre:

V = (kQ)/(R/3) = 3kQ/R

Therefore, the electric potential at a distance R/3 from the centre of the hollow metal sphere is 3kQ/R.

In summary, the electric potential at a distance R/3 from the centre of a hollow metal sphere with charge Q on its outer surface is 3kQ/R, where k is the electrostatic constant.