Gravitational Field

By the laws of physics, both the gravitational force and gravitational field intensity are connected to one common source, which is the force applied between the objects. The objects influenced by the gravitational force of another object are said to be present in their gravitational field. The magnitude by which we can have a brief idea about the force between the objects with reference to their masses is said to be gravitational field intensity. In this article, we will learn about the gravitational field due to uniform spherical shell. 

What is gravitational force? 

Gravitational force or gravity can be defined as a natural phenomenon because of which all the elements of the universe, for example, planets, satellites, stars and even light stay in their place by exerting a force on each other, i.e., gravity.

Talking about the functioning of gravity on Earth’s surface, high tides and low tides are examples of gravitational force working between Earth and Moon. The force of gravitation functioning between any two objects can be calculated as: 

F = Gm1m2 / r²   

where, 

F = force of gravitation between two objects 

G = universal gravitational constant (6.67408×10-¹¹m³/kg/s²) 

M1 = mass of the object 1 

M2 = mass of the object 2 

r = distance between the two objects 

What is the gravitational field? 

A gravitational field can be explained as a field of force created by any object in the universe around itself, which influences another object to some extent by the intensity of the field of force around them. For example,  let’s say that Moon revolves around Earth in a fixed orbit. By this we mean that both Earth and Moon are influenced by each other’s gravitational field and hence are fixed on their place. The magnitude for a gravitational field is represented by Newton per kilogram (N/kg) 

It can be calculated as: 

Gravitational field strength = (gravitational constant) × (mass of Earth)(distance of the object from Earth)²

Eg = GMr²                                                                                               

Gravitational Field Due to Uniform Spherical Shell 

While we’ve already discussed the gravitational force and the gravitational field intensity, let us take a look at some of the common phenomena that take place in this field. Gravitational field due to a uniform spherical shell is one such phenomenon that helps us understand the concept of how the gravitational field functions around an object and what impacts it creates on another object. 

Let us suppose there is a spherical shell of mass ‘m’ and radius ‘r’ with the centre as ‘O’. 

Case I 

When point ‘P’ lies outside the surface of the shell,

we have: 

Radius = r > R (r connects the centre O with the point P outside the surface) 

Mass of the shell = m 

Hence, the gravitational force exerted on the mass of the particle,

 F = GMmO / r²

Gravitational field, Eg = F / mO 

Hence, Eg = GM / r²

Case II 

When point ‘P’ lies on the surface of the spherical shell, 

 we have: 

Radius = R = r 

Mass of the shell = m 

Hence, the gravitational force exerted on the mass of the particle, 

F = GMmO / R²

Gravitational field, Eg= GMmO /R²×mO 

Hence, E= GM / R² 

Case III 

When point ‘P’ lies in the spherical shell, We have a gravitational field zero. The reason for this zero field is that the vector sum of fields due to each point on the shell cancel each other.

Conclusion

By understanding carefully all the three cases of  Uniform Spherical Shell, we found that:

•  When the point ‘P’ was positioned outside the spherical shell and the radius r > R, the magnitude of the gravitational field intensity was calculated as F = GM / r². 
• When point ‘P’ was positioned on the surface of the spherical shell and the radius r = R, the magnitude of gravitational field intensity was calculated as F = GM / R².

• Furthermore, when the point ‘P’ was positioned inside the surface of the spherical shell and the radius R > r,  the magnitude of gravitational field intensity was calculated as 0