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 find the gravitational potential at the center of the Earth (V₀) when the potential at the surface (Vₛ) is assigned a value of zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gravitational Potential**:
The gravitational potential (V) at a distance r from the center of a solid sphere of mass M and radius R is given by:
\[ V = -\frac{GM}{r} \quad \text{(for } r \geq R\text{)} \]
Inside the sphere (for \( r < R \)), the potential can be expressed as:
\[ V = -\frac{GM}{2R} + \frac{GM}{R^3} \cdot r^2 \]

2. **Setting the Potential at the Surface**:
We are given that the potential at the surface (Vₛ) is zero:
\[ Vₛ = 0 \quad \text{at } r = R \]
Therefore, we can write:
\[ 0 = -\frac{GM}{R} + \frac{GM}{R^3} \cdot R^2 \]

3. **Finding the Potential at the Center**:
To find the potential at the center (V₀), we need to evaluate the potential at \( r = 0 \):
\[ V₀ = -\frac{GM}{2R} + \frac{GM}{R^3} \cdot 0^2 \]
This simplifies to:
\[ V₀ = -\frac{GM}{2R} \]

4. **Final Result**:
Therefore, the potential at the center of the Earth when the potential at the surface is assigned a value of zero is:
\[ V₀ = -\frac{GM}{2R} \]

### Summary:
The potential at the center of the Earth, when the potential at the surface is zero, is given by:
\[ V₀ = -\frac{GM}{2R} \]