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A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The gravitational potential at a point situated at $\frac{a}{2}$ distance from the centre will be
A
$-\frac{4GM}{a}$
B
$-\frac{3GM}{a}$
C
$-\frac{2GM}{a}$
D
$-\frac{GM}{a}$
Detailed Solution
The point is inside the shell, at a distance $\frac{a}{2}$ from the centre.
Potential due to the particle at the centre: $V_1 = -\frac{GM}{a/2} = -\frac{2GM}{a}$
Potential at any point inside a spherical shell equals that on its surface: $V_2 = -\frac{GM}{a}$
Total potential: $V = V_1 + V_2 = -\frac{2GM}{a} - \frac{GM}{a}$
$V = -\frac{3GM}{a}$
Potential due to the particle at the centre: $V_1 = -\frac{GM}{a/2} = -\frac{2GM}{a}$
Potential at any point inside a spherical shell equals that on its surface: $V_2 = -\frac{GM}{a}$
Total potential: $V = V_1 + V_2 = -\frac{2GM}{a} - \frac{GM}{a}$
$V = -\frac{3GM}{a}$