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-by-Step Solution:

1. **Understanding the Gravitational Field**:
The gravitational field \( E \) at a distance \( r \) from the center of a sphere is given by the formula:
\[ E = \frac{G \cdot M}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the sphere, and \( r \) is the distance from the center of the sphere.

2. **Identifying the Point of Interest**:
In this case, we are interested in the gravitational field at the center of the sphere. Therefore, we need to set \( r = 0 \) since we are measuring the gravitational field at the center.

3. **Applying the Formula**:
Substituting \( r = 0 \) into the formula for the gravitational field:
\[ E = \frac{G \cdot M}{0^2} \]
However, this results in an undefined expression because division by zero is not possible.

4. **Understanding the Concept**:
According to the shell theorem, the gravitational field inside a uniform solid sphere is zero. This means that at any point inside the sphere, including the center, the gravitational field is zero due to the symmetrical distribution of mass around that point.

5. **Conclusion**:
Therefore, the gravitational field at the center of the uniform solid sphere is:
\[ E = 0 \]

### 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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