If potential at the surface of earth is assigned zero value , then potential at centre of earth will be (Mass = M Radius = R)

Options

A. 0
B. `-(GM)/(2R)`
C. `-(3GM)/(2R)`
D. `(3GM)/(2R)`

Answer

To solve the problem of finding the gravitational potential at the center of the Earth when the potential at the surface is assigned a value of zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that the gravitational potential at the surface of the Earth (Vs) is zero. We need to find the gravitational potential at the center of the Earth (V0).

2. **Gravitational Potential Formula**:
The gravitational potential (V) at a distance (r) from a mass (M) is given by the formula:
\[ V = -\frac{GM}{r} \]
where G is the gravitational constant.

3. **Potential at the Surface of the Earth**:
At the surface of the Earth (r = R, where R is the radius of the Earth), the potential is:
\[ V_s = -\frac{GM}{R} \]
Since we are assigning this potential a value of zero:
\[ V_s = 0 \implies -\frac{GM}{R} = 0 \]
This means we are considering the potential relative to the surface.

4. **Finding the Potential at the Center**:
To find the potential at the center of the Earth (V0), we can use the relationship between gravitational field (E) and potential (V):
\[ E = -\frac{dV}{dr} \]
For a uniform sphere, the gravitational field inside the sphere at a distance r from the center is given by:
\[ E = -\frac{GM}{R^3} r \]

5. **Integrating to Find Potential**:
We can integrate the gravitational field to find the potential:
\[ dV = -E \, dr = -\left(-\frac{GM}{R^3} r\right) dr = \frac{GM}{R^3} r \, dr \]
Integrating from the surface (r = R) to the center (r = 0):
\[ V_0 - V_s = \int_R^0 \frac{GM}{R^3} r \, dr \]

6. **Calculating the Integral**:
The integral can be computed as follows:
\[ \int r \, dr = \frac{r^2}{2} \quad \text{from } R \text{ to } 0 \]
Thus:
\[ V_0 - V_s = \frac{GM}{R^3} \left[ \frac{0^2}{2} - \frac{R^2}{2} \right] = -\frac{GM}{R^3} \cdot \frac{R^2}{2} = -\frac{GM}{2R} \]

7. **Final Potential at the Center**:
Since Vs = 0, we have:
\[ V_0 = V_s - \frac{GM}{2R} = 0 - \frac{GM}{2R} = -\frac{GM}{2R} \]

### Conclusion:
The potential at the center of the Earth is:
\[ V_0 = -\frac{GM}{2R} \]