The gravitational field due to an uniform solid sphere of mass M and radius a at the centre of the sphere is

Options

A. `(GM)/(a^(2))`
B. `(GM)/(a^(3))`
C. `(GM)/(2a)`
D. zero

Explanation

To find the gravitational field due to a uniform solid sphere of mass \( M \) and radius \( a \) at the center of the sphere, we can follow these steps:

### Step 1: Understand the Concept of Gravitational Field
The gravitational field \( \vec{E} \) at a point in space is defined as the force experienced by a unit mass placed at that point. For a mass \( M \), the gravitational field at a distance \( r \) from the center is given by the formula:
\[ \vec{E} = \frac{G M}{r^2} \]
where \( G \) is the universal gravitational constant.

### Step 2: Analyze the Situation at the Center of the Sphere
When we are at the center of a uniform solid sphere, we need to consider the contributions to the gravitational field from all parts of the sphere.

### Step 3: Use the Shell Theorem
According to the Shell Theorem, a uniform spherical shell of mass exerts no net gravitational force on a mass located inside it. Since the center of the sphere can be treated as being surrounded by a uniform shell of mass, the gravitational field at the center due to the entire sphere is zero.

### Step 4: Conclusion
Thus, the gravitational field at the center of the uniform solid sphere of mass \( M \) and radius \( a \) is:
\[ \vec{E} = 0 \]

### Final Answer
The gravitational field due to a uniform solid sphere of mass \( M \) and radius \( a \) at the center of the sphere is \( 0 \).

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