Question

A particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a$. The magnitude of the gravitational potential at a point situated at $a/2$ distance from the centre, will be :

• A. $\frac{GM}{a}$
• B. $\frac{2GM}{a}$
• C. $\frac{3GM}{a}$
• D. $\frac{4GM}{a}$

Answer 1

The Correct Option is C

To determine the gravitational potential at a point situated at a distance \( \frac{a}{2} \) from the center of the spherical shell, we need to consider the contributions of gravitational potential from both the particle at the center and the spherical shell.

1. Gravitational Potential due to the Central Mass:
   • The gravitational potential \( V_c \) at a distance \( r \) from a point mass \( M \) is given by:
   \[ V_c = -\frac{GM}{r} \]
   • For the mass at the center \( M \) and distance \( r = \frac{a}{2} \), the potential is:
   \[ V_c = -\frac{GM}{\frac{a}{2}} = -\frac{2GM}{a} \]
2. Gravitational Potential due to the Spherical Shell:
   • According to the shell theorem, the gravitational potential inside a spherical shell of mass \( M \) and radius \( a \) at any point inside the shell is constant and equal to the potential at the surface.
   • The potential \( V_s \) at any point inside (including the center) is given by:
   \[ V_s = -\frac{GM}{a} \]
3. Total Gravitational Potential at the Distance \( \frac{a}{2} \):
   • The total potential \( V \) at distance \( \frac{a}{2} \) from the center is the algebraic sum of the potentials due to both the central mass and the shell.
   \[ V = V_c + V_s = -\frac{2GM}{a} - \frac{GM}{a} \]
   \[ V = -\frac{3GM}{a} \]
4. Conclusion:
   • Since the potential is asked in terms of magnitude, it will be \( \frac{3GM}{a} \).

Thus, the correct answer is \(\frac{3GM}{a}\).