Gravitational Intensity due to a solid sphere:

Gravitational intensity at a point in a gravitational field is defined as the attractive force experienced by a unit mass placed at that point. READ IN DETAIL.

(i)At a point outside the sphere:

Gravitational potential at a point outside the sphere of radius \( a \) at a distance \( r \) \( (r>a) \) from the centre of the sphere is given by, \( V=-\frac{GM}{r} \).

Gravitational Potential Due To A Solid Homogeneous Sphere At A Point (i) Outside, (ii) On The Surface, And (iii) Inside A Point Of The Sphere. READ IN DETAIL.

Now, gravitational Intensity

\( E= -\frac{\,dv}{\,dr}=-\frac{d}{\,dr}(-\frac{GM}{r}) \\=GM\frac{d}{\,dr}(\frac{1}{r}) \\=-\frac{GM}{r^2} \).

(ii) At a point on the surface of the sphere:

Gravitational potential at a point on the surface of the solid sphere is

\( \displaystyle{V=-\left(\frac{GM}{r}\right)_r=a} \)

Gravitational Intensity is

\( E=-\frac{\,dv}{\,dr}=-\frac{d}{\,dr} {-\left(\frac{GM}{r}\right)_{r=a}}\\= \left(-\frac{GM}{r^2}\right)_{r=a}\\=-\frac{GM}{a^2} \)

(iii) At a point inside the sphere:

Lets consider a point \( P \) is situated inside the sphere at a distance \( r \) from the centre \( O \) of the sphere. \( (r<a) \).

Then the potential at \( P \) due to this sphere is given by,

\( \displaystyle{V=-GM\frac{3a^2-r^2}{2a^3}} \)

Gravitational Intensity at \( P \) is

\( E=-\frac{\,dv}{\,dr}=\frac{d}{\,dr}\left( GM\frac{3a^2-r^2}{2a^3} \right)\\=\frac{GM}{2a^3}\cdot\frac{d}{\,dr}(3a^2-r^2)\\=\frac{GM}{2a^3}(-2r^2)\\=-\frac{GM}{a^3}\cdot{r} \)

Here the gravitational intensity at a point inside the solid sphere is proportional to its distance from the centre.

Graphical Representation: