Gravitational Field Due to Spherical Shell

Gravitational Field Due to Spherical Shell

Here we will discuss thoroughly the Gravitational Field Due to Spherical Shell. We will also discuss the situation when the test point is outside of the shell as well as when it is inside of the shell. Here we will use the final expression for the gravitational field due to uniform ring as a ready reference. So if you are not familiar with that expression you can go back and start from there.

You can also start your journey with the Laws of Gravitation.

Gravitational Field due to Spherical Shell

As already mentioned, we define the intensity of the gravitational field \vec{E} at any point as the gravitational force per unit mass.  Let us consider a uniform spherical shell of mass M. The center of mass is located at point O. We have to calculate the field due to this spherical shell at a test point P, as shown in the figure below. We denote the distance between O and P as r. First, to calculate the field at point P due to the mass element dM (green region), we will use the expression of the gravitational field derived for the uniform circular ring. The field due to this ring at point P is

gravitational field due to spherical shell

    \[ dE=\frac{GdM}{Z^2}Cos(\alpha)=\frac{GM Sin(\theta)d\theta Cos(\alpha))}{2z^2}\]

Please remember that the field is along the direction of \vec{PO}.  From the triangle OAP,

    \[ z^2=a^2+r^2 - 2ar Cos(\theta)\]

Now, taking differential for both sides,

    \[ 2z dz=2ar Sin(\theta) d\theta\]

    \[ Sin(\theta)d\theta= \frac{zdz}{ar}\]

Again from the same triangle,

    \[ a^2=z^2+r^2 - 2zr Cos(\alpha)\]

Or

    \[ Cos(\alpha)= \frac{z^2+r^2-a^2}{2zr}\]

Using these two equations, we arrive at

    \[ dE= \frac{GM}{4ar^2}\left(1-\frac{a^2-r^2}{z^2}\right) dz\]

    \[ E= \frac{GM}{4ar^2}\left \int \left ( 1-\frac{a^2-r^2}{z^2} \right) dz \right\]

Test Point P is Outside of the Shell

Here z varies from r-a to r+a. Therefore, the field due to spherical shell at P is 

    \[ E= \frac{GM}{4ar^2}\left (z+\frac{a^2-r^2}{z}\right)_{r-a}^{r+a}=\frac{GM}{r^2}\]

The direction of this field is towards the center of the shell. Thus,

    \[ \boxed{\vec E=-\frac{GM}{r^2}\hat{r}}\]

For an external test point, shell behaves like a point mass placed at the center of mass of the shell.

Test Point P is Inside of the Shell

Here z varies from a-r to a+r. Therefore the field due to the shell is 

    \[ E= \frac{GM}{4ar^2}\left (z+\frac{a^2-r^2}{z}\right)_{a-r}^{a+r}=0\]

Field inside a spherical shell is always zero.

In the next article we will use this result to discuss about the Gravitational Filed due to a Uniform Solid Sphere.

Gravitational Field due to Spherical Shell as a function of distance from the center of the shell

In the diagram above, we have plotted the gravitational field due to a spherical shell as a function of distance from the center of the shell.

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