Gravitational potential at the centre of a uniform solid sphere of mass ‘M’ and radius ‘R’ is (if potential due to the sphere at the infinity is GM /R.)

To determine the gravitational potential at the center of a uniform solid sphere, we need to consider how gravitational potential is defined and how it behaves within and outside a sphere. The gravitational potential \( V \) at a point in space due to a mass \( M \) is given by the formula:

V = -\frac{GM}{r}

where \( G \) is the gravitational constant, \( M \) is the mass of the sphere, and \( r \) is the distance from the center of the mass to the point where the potential is being calculated. However, this formula applies outside the mass. Inside a uniform solid sphere, the situation is a bit different.

Understanding Gravitational Potential Inside the Sphere

Inside a uniform solid sphere, the gravitational potential does not vary linearly with distance from the center. Instead, we can derive the potential at the center by considering the contributions from all the mass elements of the sphere.

Step-by-Step Derivation

• Potential at Infinity: The gravitational potential at infinity is defined as zero. Thus, we can set our reference point here.
• Potential Inside the Sphere: For a point inside the sphere, the gravitational potential can be calculated by integrating the contributions from each mass element. However, a simpler approach is to recognize that the potential inside a uniform sphere is constant.
• Using the Shell Theorem: According to the shell theorem, a uniform spherical shell of mass exerts no net gravitational force on a particle located inside it. Therefore, the gravitational potential inside the sphere is the same as that at the surface of the sphere.

The gravitational potential at the surface of the sphere (at radius \( R \)) can be calculated using the formula:

V(R) = -\frac{GM}{R}

Since the potential is constant throughout the interior of the sphere, the gravitational potential at the center (where \( r = 0 \)) is equal to the potential at the surface:

V(center) = -\frac{GM}{R}

Final Result

Thus, the gravitational potential at the center of a uniform solid sphere of mass \( M \) and radius \( R \) is:

V(center) = -\frac{GM}{R}

This result illustrates how gravitational potential behaves within a uniform solid sphere, highlighting the unique properties of gravitational fields and potentials in such symmetric mass distributions. Understanding these concepts is crucial for further studies in gravitational physics and astrophysics.