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Last updated on December 2nd, 2019 at 03:50 am

The main interest of this article is to calculate ** Gravitational Potential due to Solid Sphere**. However, if you are interested in a detailed study of

*, you can start here. In our previous article, we have discussed how to calculate*

**Gravitation****. We will use that knowledge to solve this problem. Again, the main idea will be to calculate the amount of work done to move a unit mass system from a point of reference to the field of influence of the source mass M.**

*Gravitational Potential due to a Spherical shell*## Gravitational Potential due to Solid Sphere

Let us consider a uniform sphere of mass M and radius a. The center of mass is located at point O. We have to calculate the potential due to this solid sphere at a test point P, as shown in the figure below. We denote the distance between O and P as . By definition, the potential is the work done to move a unit mass from a point of reference, here infinity to the test point .

We will first draw two concentric spherical shells of radius and . The volume enclosed by this thick shell of interest is . We can easily calculate the mass due to this thick spherical shell as

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Now we can use the final expression for ** gravitational potential due to a spherical shell** to calculate the

**at a point either outside or inside of the sphere.**

*gravitational potential due to a solid sphere*## Test Point P is Outside of the Sphere

The most trivial situation is that the test point P is outside of the sphere. We use our ready-made equation from earlier exercise to calculate the potential at this point.

Thus the potential due to the solid sphere will be

Again we note that the ** gravitational potential due to a solid sphere** at an external point is identical to the

**M situated at the center of the sphere.**

*gravitational potential due to a point mass*## Related Posts

## Test Point P is Inside of the Sphere

Here the situation is more complex than the earlier one, however, we can divide the sphere into two parts. The first one which is a sphere of radius and the second one is a thick spherical shell with the inner radius of and outer radius of . Let us assume that the mass of the inner sphere is M’, then

and the potential at a test point, P due to this sphere is

Now we have to calculate potential at point P due to the spherical shell of inner radius and outer radius . For that, we will divide the shell into several concentric shells. Let us consider one such shell with an inner radius of and outer radius . The corresponding mass of the shell is

The potential at an internal point P due to this shell is

Thus the potential due to this part is

and the total ** gravitational potential due to a solid sphere** at an internal point is

## Gravitational Potential due to Cavity

In the following section, we will calculate the gravitational potential due to a spherical cavity. Let us consider a solid sphere of radius a. We create a spherical cavity as shown in the diagram below. The radius of the spherical cavity is . Taking gravitational potential at is zero, calculate the potential at the center of the cavity.

Let , and are the potential due to the solid sphere, the sphere with a cavity and the cavity respectively. We have to calculate at point P.

Point P is inside of the sphere thus the gravitational potential at point P due to the solid sphere is

Point P is at the center of the cavity thus the gravitational potential at point P due to the cavity is

Here is the mass of the cavity and is . Thus

Now, the total gravitational potential at point P is the sum of the potential due to the sphere with a cavity and the potential due to the cavity or . Thus

In the diagram below, we have plotted the gravitational potential due to a solid sphere as a function of distance from the center of the sphere. Please be careful about the y-axis in the plot.