The variation of electrostatic potential with radial distance r from centre of positively charged metallic thin shell of radius R is given by graph

Options

A.
B. `##NEET_RE_20_PHY_33_O02##`
C. `##NEET_RE_20_PHY_33_O03##`
D. `##NEET_RE_20_PHY_33_O04##`

Answer

To solve the problem of the variation of electrostatic potential with radial distance \( r \) from the center of a positively charged metallic thin shell of radius \( R \), we will analyze the potential in three regions: inside the shell, on the surface, and outside the shell.

### Step-by-Step Solution:

1. **Understanding the Electric Field Inside the Shell:**
- For a metallic shell, the electric field \( E \) inside the shell (for \( r < R \)) is zero. This is due to the property of conductors in electrostatic equilibrium, where charges reside on the surface and do not create an electric field inside.

**Hint:** Recall that for a conductor in electrostatic equilibrium, the electric field inside is zero.

2. **Potential Inside the Shell:**
- Since the electric field \( E = 0 \) inside the shell, the potential \( V \) remains constant throughout this region. The potential inside the shell is equal to the potential at the surface.

**Hint:** Remember that if \( E = 0 \), then \( dV = -E \, dr \) implies \( dV = 0 \).

3. **Calculating the Potential on the Surface:**
- The potential \( V \) at the surface of the shell (for \( r = R \)) can be calculated using the formula for the potential due to a point charge:
\[ V(R) = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{R} \]
- Here, \( Q \) is the total charge on the shell.

**Hint:** Use the formula for the potential due to a point charge to find the potential at the surface.

4. **Potential Outside the Shell:**
- For \( r > R \), the shell behaves like a point charge located at its center. The potential at a distance \( r \) from the center is given by:
\[ V(r) = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{r} \]
- This shows that the potential decreases as \( r \) increases.

**Hint:** Remember that outside a charged shell, the potential behaves like that of a point charge.

5. **Summary of Potential Variation:**
- **For \( r < R \)**: The potential \( V \) is constant and equal to \( V(R) \).
- **For \( r = R \)**: The potential is \( V(R) = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{R} \).
- **For \( r > R \)**: The potential \( V \) decreases as \( \frac{1}{r} \).

### Final Answer:
The variation of electrostatic potential \( V \) with radial distance \( r \) from the center of the positively charged metallic thin shell is:
- Constant for \( r < R \),
- Equal to \( \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{R} \) at \( r = R \),
- Decreasing as \( \frac{1}{r} \) for \( r > R \).