The dependence of electric potential V on the distance 'r' from the centre of a charged spherical shell is shown by :

Options

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D.

Answer

To solve the problem of determining the dependence of electric potential \( V \) on the distance \( r \) from the center of a charged spherical shell, we will follow these steps:

### Step 1: Understand the Concept of Electric Potential
The electric 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 without any acceleration.

### Step 2: Analyze the Electric Field Inside and Outside the Spherical Shell
For a charged spherical shell:
- **Inside the shell**: The electric field \( E \) is zero. Therefore, the potential \( V \) remains constant throughout the interior of the shell.
- **Outside the shell**: The electric field \( E \) behaves as if all the charge were concentrated at the center of the shell. The electric field at a distance \( r \) from the center (where \( r > R \), \( R \) being the radius of the shell) is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k = \frac{1}{4\pi\epsilon_0} \) and \( Q \) is the total charge on the shell.

### Step 3: Calculate the Electric Potential Outside the Shell
Using the relation between electric field and potential:
\[ dV = -E \, dr \]
Integrating from infinity to a point \( r \) outside the shell:
\[ V(r) - V(\infty) = -\int_{\infty}^{r} E \, dr = -\int_{\infty}^{r} \frac{kQ}{r^2} \, dr \]
Calculating the integral:
\[ V(r) - 0 = -\left[-\frac{kQ}{r}\right]_{\infty}^{r} = \frac{kQ}{r} \]
Thus, for \( r > R \):
\[ V(r) = \frac{kQ}{r} \]

### Step 4: Calculate the Electric Potential Inside the Shell
Since the electric field inside the shell is zero:
\[ E = 0 \implies dV = 0 \]
Thus, the potential \( V \) inside the shell is constant and equal to the potential at the surface of the shell:
\[ V = V(R) = \frac{kQ}{R} \]

### Step 5: Summarize the Results
- For \( r < R \) (inside the shell):
\[ V = \frac{kQ}{R} \quad \text{(constant)} \]
- For \( r > R \) (outside the shell):
\[ V = \frac{kQ}{r} \quad \text{(decreasing with increasing } r\text{)} \]

### Conclusion
The graph of \( V \) versus \( r \) will show a constant value \( \frac{kQ}{R} \) for \( r < R \) and a hyperbolic decrease for \( r > R \). Therefore, the correct option for the dependence of electric potential \( V \) on the distance \( r \) from the center of a charged spherical shell is that \( V \) is constant inside the shell and decreases as \( r \) increases outside the shell.