Gravitational Potential at a Point Inside a Spherical Shell

When a particle of mass M is situated at the center of a spherical shell of the same mass and radius a, we need to find the magnitude of the gravitational potential at a point situated at a/2 distance from the center.

To calculate the gravitational potential at this point, we can consider the spherical shell as a collection of concentric spherical shells of infinitesimal thickness. The gravitational potential due to each infinitesimal shell is given by the formula:

ΔV = -G(dm)/r

where ΔV is the gravitational potential due to the infinitesimal mass element dm, G is the gravitational constant, and r is the distance between the mass element and the point where we want to calculate the potential.

Gravitational Potential Due to the Shell

Let's consider a spherical shell of radius r1 and thickness dr. The mass of this shell can be calculated using the formula:

dm = (M * 4πr1² * dr) / (4/3πa³)

The distance between the mass element dm and the point at a/2 distance from the center can be approximated as r2 = a/2.

Using the formula for gravitational potential, we can calculate the potential due to this shell as:

ΔV = -G(dm)/r2
= -G(M * 4πr1² * dr) / (4/3πa³) / (a/2)
= -2G(M * r1² * dr) / (a²)

Gravitational Potential due to the Entire Shell

To find the total gravitational potential at the point, we need to integrate the contributions from all the infinitesimal shells from r = 0 to r = a.

V = ∫ΔV = ∫[-2G(M * r1² * dr) / (a²)]
= -2GM / a² ∫[r1² * dr]
= -2GM / a² ∫[r1² * dr]
= -2GM / a² * [(1/3)r1³] (from 0 to a)
= -2GM / a² * (1/3)a³
= -2GMa / 3a²
= -2GM / 3a

Final Answer

Therefore, the magnitude of the gravitational potential at a point situated at a/2 distance from the center is given by -2GM / 3a.