A hollow sphere of radius r is put inside another hollow sphere of radius R. The charges on the two are +Q and -q as shown in the figure. A point P is located at a distance x from the common centre such that `r lt x lt R`. The potential at the point P is

Options

A. `1/(4pi epsi_(0)) ((Q-q)/x)`
B. `1/(4pi epsi_(0)) (Q/R-q/r)`
C. `1/(4pi epsi_(0)) (Q/R-q/x)`
D. `1/(4pi epsi_(0)) (q/r-Q/x)`

Answer

To find the electric potential at point P located at a distance \( x \) from the common center of two concentric hollow spheres with charges \( +Q \) and \( -q \), we will follow these steps:

### Step 1: Understand the Configuration
We have two hollow spheres:
- The inner sphere has a radius \( r \) and charge \( +Q \).
- The outer sphere has a radius \( R \) and charge \( -q \).
- The point P is located at a distance \( x \) from the center, where \( r < x < R \).

### Step 2: Determine the Potential Due to Each Sphere
1. **Potential due to the Outer Sphere (V1)**:
- For a hollow sphere, the electric field inside the sphere is zero. Therefore, the potential inside the outer sphere (at point P) is constant and equal to the potential at its surface.
- The potential \( V_1 \) due to the outer sphere at the surface is given by:
\[ V_1 = \frac{kQ}{R} \]
where \( k \) is the Coulomb's constant.

2. **Potential due to the Inner Sphere (V2)**:
- The potential inside a hollow charged sphere is also constant and equal to the potential at its surface.
- The potential \( V_2 \) due to the inner sphere at the surface is given by:
\[ V_2 = \frac{k(-q)}{r} \]
where \( -q \) is the charge on the inner sphere.

### Step 3: Calculate the Total Potential at Point P
- The total potential \( V_P \) at point P is the sum of the potentials due to both spheres:
\[ V_P = V_1 + V_2 \]
Substituting the expressions for \( V_1 \) and \( V_2 \):
\[ V_P = \frac{kQ}{R} + \frac{k(-q)}{r} \]
Simplifying this gives:
\[ V_P = \frac{kQ}{R} - \frac{kq}{r} \]

### Final Expression for the Potential at Point P
Thus, the potential at point P is given by:
\[ V_P = \frac{kQ}{R} - \frac{kq}{r} \]