In case of earth

Options

A. gravitational field is zero both at centre and infinity
B. gravitational potential is zero, both at centre and infinity
C. gravitational potential is same both at centre and infinity but not zero
D. gravitational potential is minimum at the centre

Answer

To solve the question regarding the gravitational field and gravitational potential in the case of Earth, we will analyze both concepts step by step.

### Step-by-Step Solution:

**Step 1: Understanding Gravitational Field at Different Points**
- The gravitational field (g) at a point is defined as the force experienced by a unit mass placed at that point.
- At the center of the Earth, the gravitational field is zero. This is because the mass of the Earth exerts equal gravitational forces in all directions, resulting in a net force of zero.
- At a depth \(d\) below the surface of the Earth, the gravitational field can be calculated considering only the mass of the Earth that is at a radius less than \(r\). The mass outside this radius does not contribute to the gravitational field at that point.

**Hint for Step 1:** Remember that inside a uniform spherical shell, the gravitational field is zero.

**Step 2: Gravitational Field at Infinity**
- The gravitational field at infinity is also zero. This is because as the distance \(r\) from the mass increases, the gravitational field \(g\) is given by the formula:
\[ g = \frac{GM}{r^2} \]
As \(r\) approaches infinity, \(g\) approaches zero.

**Hint for Step 2:** Think about how gravitational force diminishes with distance.

**Step 3: Gravitational Potential at the Center of the Earth**
- Gravitational potential (V) is defined as the work done in bringing a unit mass from infinity to that point.
- The formula for gravitational potential at a distance \(r\) from the center of a mass \(M\) is:
\[ V = -\frac{GM}{r} \]
- At the center of the Earth, the gravitational potential can be derived from integrating the gravitational field. It is given by:
\[ V = -\frac{3GM}{2R} \]
where \(R\) is the radius of the Earth. This value is negative, indicating that it is a bound system.

**Hint for Step 3:** Remember that gravitational potential is always negative and reaches its minimum at the center.

**Step 4: Conclusion on Gravitational Potential**
- Since the gravitational potential is negative and reaches its minimum value at the center of the Earth, we conclude that the gravitational potential is minimum at the center.
- Therefore, the correct options are:
- Gravitational field is zero at the center and at infinity.
- Gravitational potential is minimum at the center.

**Final Answer:**
- Gravitational field is zero at the center and at infinity (Option A).
- Gravitational potential is minimum at the center (Option D).