P is a point at a distance r from the centre of a spherical shell of mass M and radius a, where `r lt a` The gravitational potential at P is

Options

A. `-(GM)/r`
B. `-(GM)/a`
C. `-GM(r)/a^2`
D. `-GM((a-r)/a^2)`

Answer

To find the gravitational potential at point P, which is located at a distance \( r \) from the center of a spherical shell of mass \( M \) and radius \( a \), where \( r < a \), we can follow these steps:

### Step 1: Understand the Concept of Gravitational Potential
Gravitational potential \( U \) at a point in space due to a mass \( M \) is given by the formula:
\[ U = -\frac{G M}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass creating the gravitational field, and \( r \) is the distance from the mass to the point where the potential is being calculated.

### Step 2: Apply the Shell Theorem
According to the shell theorem, the gravitational potential inside a uniform spherical shell of mass is constant and equal to the potential at the surface of the shell. Therefore, for any point inside the shell (where \( r < a \)), the gravitational potential \( U \) is given by:
\[ U = -\frac{G M}{a} \]
where \( a \) is the radius of the shell.

### Step 3: Write the Final Expression for Gravitational Potential at Point P
Since point P is inside the shell, we can directly use the expression derived from the shell theorem:
\[ U = -\frac{G M}{a} \]

### Conclusion
Thus, the gravitational potential at point P, located at a distance \( r \) from the center of the spherical shell, is:
\[ U = -\frac{G M}{a} \]