The gravitational potential due to the earth is minimum at

Options

A. the centre
B. the surface
C. a distnace equal to 100 times the radius of the earth
D. infinite distance

Answer

To determine where the gravitational potential due to the Earth is minimum, we can analyze the gravitational potential formula and its behavior at different locations relative to the Earth.

### Step-by-Step Solution:

1. **Understand Gravitational Potential**:
The gravitational potential \( V \) due to a mass \( M \) (in this case, the Earth) at a distance \( r \) from its center is given by the formula:
\[ V = -\frac{GM}{r} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Evaluate at Different Points**:
We need to evaluate the gravitational potential at various points:
- **At the Surface of the Earth**:
At the surface, \( r = R \) (the radius of the Earth).
\[ V = -\frac{GM}{R} \]
- **At the Center of the Earth**:
At the center, \( r = 0 \). The formula becomes problematic as it leads to division by zero. However, conceptually, the gravitational potential at the center can be considered to approach negative infinity:
\[ V = -\frac{GM}{0} \rightarrow -\infty \]
- **At a Distance of 100 Times the Radius of the Earth**:
At this distance, \( r = 100R \).
\[ V = -\frac{GM}{100R} \]
- **At Infinite Distance**:
As \( r \) approaches infinity, the gravitational potential approaches zero:
\[ V = -\frac{GM}{\infty} = 0 \]

3. **Compare Values**:
- At the surface: \( V = -\frac{GM}{R} \)
- At the center: \( V = -\infty \)
- At 100R: \( V = -\frac{GM}{100R} \)
- At infinity: \( V = 0 \)

4. **Determine Minimum Value**:
The lowest (most negative) value of gravitational potential occurs at the center of the Earth, where it approaches negative infinity. Therefore, the gravitational potential due to the Earth is minimum at the center.

### Conclusion:
The gravitational potential due to the Earth is minimum at the **center of the Earth**.