The electrostatic potential of a uniformly charged thin spherical shell of charge Q and radius R at a distance `r` from the centre

Options

A. `(Q)/(4 pi epsilon_(0)r)` for points outside and `(Q)/(4pi epsilon_(0)R)` for points inside the shell
B. `(Q)/(4pi epsilon_(0)e)` for both points inside nad outside the shell
C. zero for points outside and `(Q)/(4pi epsilon_(0)r)` for points inside the shell
D. zero for both points inside and outside the shell

Answer

To find the electrostatic potential \( V \) of a uniformly charged thin spherical shell of charge \( Q \) and radius \( R \) at a distance \( r \) from the center, we can analyze two cases: when the point is inside the shell and when it is outside the shell.

### Step 1: Understand the Concept of Electrostatic Potential
Electrostatic potential \( V \) at a point in space due to a charge distribution is defined as the work done in bringing a unit positive charge from infinity to that point.

### Step 2: Analyze the Case for Points Inside the Shell (\( r < R \))
For a uniformly charged thin spherical shell, the electric field inside the shell is zero. This can be derived from Gauss's law, which states that the electric field inside a uniformly charged shell is zero because the contributions from all parts of the shell cancel out.

Since the electric field \( E = 0 \) inside the shell, the potential \( V \) remains constant throughout the interior of the shell. The potential inside the shell is equal to the potential at the surface of the shell.

### Step 3: Calculate the Potential at the Surface of the Shell
The potential at the surface of the shell (at distance \( R \)) can be calculated using the formula for the potential due to a point charge:
\[ V(R) = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant.

### Step 4: Conclude for Points Inside the Shell
Thus, for any point inside the shell (\( r < R \)):
\[ V = \frac{kQ}{R} \]

### Step 5: Analyze the Case for Points Outside the Shell (\( r \geq R \))
For points outside the shell, the shell behaves like a point charge located at its center. Therefore, we can use the same formula for the potential due to a point charge:
\[ V(r) = \frac{kQ}{r} \]
where \( r \) is the distance from the center of the shell.

### Step 6: Conclude for Points Outside the Shell
Thus, for any point outside the shell (\( r \geq R \)):
\[ V = \frac{kQ}{r} \]

### Final Summary
- For \( r < R \) (inside the shell):
\[ V = \frac{kQ}{R} \]
- For \( r \geq R \) (outside the shell):
\[ V = \frac{kQ}{r} \]