What is Gravitational Field?

Gravitational Field is a region of space where a mass such as a planet, moon, or star causes objects to experience a force of attraction.

The source mass and test mass interact with each other through a gravitational field in a non-contact force. You can think of the gravitational force as a “command” and the gravitational field as the dialogue or speech used to give the command.

Table of Contents:

• What is Gravitational Field Intensity?
• Gravitational Field Intensity of a Point Mass
• Ring
• Gravitational Field Intensity due to Uniform Spherical Shell
• Gravitational Field Intensity due to Uniform Solid Sphere
• Solved Examples

What is Gravitational Field Intensity?

Gravitational Field Intensity is the strength of the gravitational field at a given point in space. It is measured in units of acceleration, such as meters per second squared (m/s²).

The strength of the gravitational field is known as gravitational field intensity. It is the gravitational force acting on a unit test mass.

E = F/m

Either

\begin{array}{l}E_g = \left[-\frac{GM}{r^2}\right] \hat{r}\end{array}

⇒ Gravitational Field Intensity: $$\mathbf{E_g = \left[-\frac{GM}{r^2}\right]\hat{r}}$$

The position vector of the test mass from the source mass is represented by $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$, where $\hat{r}$ is the unit vector along the radial direction.

The gravitational field intensity is determined solely by the mass of the source and the distance between a unit test mass and the source mass.

The unit of gravitational field intensity is N/kg.

The dimensional formula is given by [M⁰L¹T^-2].

The dimensional formula of gravitational field intensity is the same as the acceleration due to gravity (which is the more preferable term from the perspective of gravitation).

The superposition principle can be applied to gravitational field intensities,

$E=\sum _{i=1}^n{E}_i$

Where $$E_1, E_2, E_3, \ldots, E_n$$ are the gravitational field intensities at a point due to n particles in a system.

In a system, the mass is typically distributed in two different ways:

Discrete Mass Distribution

Continuous Mass Distribution

For a discrete mass distribution: \(\vec{E} = \sum_{i=1}^{n}E_i\)

For a continuous mass distribution: ${\int }_i^f\overrightarrow{E}dE$

The gravitational field formula for the intensity due to an elementary mass dm is expressed as dE.

$g=\frac{F}{m}$

F = gravitational force and m = mass of the object.

⇒ Also Read:

Kepler’s Laws of Planetary Motion

Gravitational Potential Energy

Gravitational Field Strength of a Point Mass

The gravitational intensity of a point mass M at a distance ‘r’ is given by

\begin{array}{l}E_g = \left[-\frac{GM}{r^2}\right]\hat{r}\end{array}

Gravitational Field Intensity of a Ring

The gravitational field at a distance x along the axis of a ring of mass M and radius ‘a’ can be determined as follows:

Consider a small length element along the circumferential length of the ring, with a mass dm, the field intensity due to this length element is given by;

DE = Gdm/r²

Only the horizontal components of the fields remain due to the symmetry of the ring, and they add up to form a resultant vector.

\begin{array}{l}E = \int_{0}^{2\pi} \frac{Gdm}{r^{2}} \cos{\alpha} ,d\alpha\end{array}

Since $$cos\alpha = \frac{x}{\sqrt{x^2 + a^2}}$$

\begin{array}{l}E=\frac{GMx}{{{\left( {{x}^{2}}+{{a}^{2}} \right)}^{\frac{3}{2}}}}\end{array}

Gravitational Field of a Uniform Spherical Shell

Consider a thin uniform spherical shell of radius ‘R’, with mass ‘M’, situated in a space. A 3D object divides the space into three parts:

Inside the spherical shell.

On the surface of the spherical shell.

Outside the spherical shell.

Our goal is to determine the gravitational field intensity in each of these three regions.

Outside the Spherical Shell

Draw an imaginary spherical shell such that point ‘P’ lies on its surface, where ‘P’ is a unit test mass at a distance ‘r’ from the centre of the shell.

We can conclude that the gravitational field intensity at a point within an imaginary sphere, with source mass M and distance of separation ‘r’, depends solely on M and ‘r’.

$E=-\frac{GM}{r^2}$

$E\propto -1/r^2$

On the Surface of a Spherical Shell

The gravitational field intensity on the surface of a spherical shell at a point ‘P’ a distance ‘r’ from the centre is given by r = R, where R is the radius of the shell.

$E=-\frac{GM}{r^2}$

E = Constant

⇒ Check: Acceleration due to Gravity

Inside the Spherical Shell

Draw an imaginary spherical shell about point ‘P’. If we consider a point inside the shell, the entire mass of the shell lies above the point and the source mass within this imaginary sphere will be zero.

If the source mass is equal to zero, then the gravitational field intensity is also equal to zero.

E = 0

Conclusions:

Position of Point ‘P’	Gravitational Field Intensity
Inside the spherical shell (r < R)	E = 0
On the surface of the spherical shell (r = R),	$E=-GM/R^2$
Outside the spherical shell (r > R)	$E=\frac{-GM}{r^2}$

Gravitational Field of a Uniform Solid Sphere

Consider a uniform solid sphere of radius ‘R’ and mass ‘M’. Let us determine the value of gravitational field intensity in the following three regions:

Inside the solid sphere.

On the surface of a solid sphere.

Outside the solid sphere.

Outside the Solid Sphere

Consider an imaginary sphere about point P, which is at a distance r from the centre of a solid sphere, and encloses the entire mass M. This will help to find the gravitational field intensity at point P.

$E=\frac{-GM}{r^2}$

$E\propto -1/r^2$

On the Surface of a Solid Sphere

To calculate the gravitational field intensity at a point ‘P’ located on the surface of a solid sphere.

The distance to the point on the surface is r = R.

Then, $E=\mathrm{Constant}\leftarrow -GM/R^2$

Inside the Solid Sphere

If we draw an imaginary sphere of radius ‘r’ about the point ‘P’ situated inside a uniform solid sphere, the mass ’m’ present within this imaginary sphere can be used to calculate the gravitational influence at that point.

For a volume of $$(4/3)\pi R^3$$, the mass present is M, for a volume of $$(4/3)\pi r^3$$, the mass present is m.

As the density of the solid sphere remains constant throughout,

Therefore, m = M × (r³/R³)

The gravitational field intensity at point ‘P’ inside a solid sphere at a distance ‘r’ from the centre of the sphere is given by:

$$E = \frac{-Gm}{r^2}$$

By substituting the value of m in the equation above, we get

$$E = -\frac{GM}{R^3}$$

$E\propto -r$

Conclusions:

Position of Point ‘P’	Gravitational Field Intensity
Inside the uniform solid sphere (r < R)	$E=-\frac{GM}{r^3}$
On the surface of the uniform solid sphere (r = R)	$E=-GM/R^2$
Outside the uniform solid sphere (r > R)	$E=-GM/r^2$

Solved Examples

Example 1: Determine the gravitational field if the gravitational force and mass of a substance are 10N and 5kg.

Solution:

The given parameters are: F = 10 N and m = 5 kg

The formula for gravitational field intensity is given by:

g = F/m = 10/5 = 2 N/kg

Example 2: Calculate the gravitational field if the mass and force of a substance are given as 6 kg and 36 N, respectively.

Solution:

The given parameters are:

• F = 36 N
• m = 6 kg

g = 6 N/kg

Frequently Asked Questions about Gravitational Field Intensity

What is the unit of measure for the intensity of the gravitational field?

N/kg

What is a Gravitational Field?

A gravitational field is a physical field generated by a massive object, such as a planet or star, that exerts a force on other objects in its vicinity. It is responsible for the attraction between objects with mass, such as the Earth and the Moon.

Gravitational field intensity is a measure of the strength of the gravitational field at a given point in space.

The gravitational field intensity at a point is defined as the force experienced by a unit mass placed at that point.

Gravitational potential is a measure of the potential energy of an object in a gravitational field, which is equal to the work done by the field in bringing the object from infinity to its present position.

The gravitational potential at a point in the gravitational field of a body is defined as the amount of work done in bringing a body of unit mass from infinity to that point in the gravitational field.