Electric potential inside hollow charged sphere of radius .r. is ______ `[k = (1)/(4pi epsilon_(0))]`

Options

A. zero
B. `k(q)/(r )`
C. `k(q)/(r^(2))`
D. `k(q^(2))/(r )`

Answer

To find the electric potential inside a hollow charged sphere of radius \( r \), we can follow these steps:

### Step 1: Understand the configuration
We have a hollow charged sphere with a uniform charge distributed over its outer surface. The sphere has a radius \( r \).

### Step 2: Recognize the properties of electric fields in hollow spheres
According to Gauss's law, the electric field inside a hollow charged sphere is zero. This is because the charge resides only on the outer surface, and by symmetry, any point inside the hollow region experiences no net electric field.

### Step 3: Relate electric field to electric potential
The electric potential \( V \) is related to the electric field \( E \) by the equation:
\[ V = -\int E \cdot dr \]
Since the electric field \( E \) inside the hollow sphere is zero, the integral simplifies.

### Step 4: Calculate the electric potential inside the sphere
Since the electric field \( E \) is zero inside the hollow sphere, the potential \( V \) remains constant throughout the hollow region. The potential inside the sphere is equal to the potential at the surface of the sphere.

### Step 5: Calculate the potential at the surface
The electric potential \( V \) at the surface of the sphere (at radius \( r \)) due to the charge \( Q \) on the sphere is given by:
\[ V = \frac{kQ}{r} \]
where \( k = \frac{1}{4\pi \epsilon_0} \).

### Step 6: Conclude the potential inside the hollow sphere
Since the potential is constant inside the hollow sphere and equal to the potential at the surface, we conclude:
\[ V_{\text{inside}} = V_{\text{surface}} = \frac{kQ}{r} \]

Thus, the electric potential inside the hollow charged sphere of radius \( r \) is:
\[ \frac{kQ}{r} \]