Gravitational Field Intensity | JEE, NEET & CBSE Board Class 11 Exams Notes

What is Gravitational Field ?

For a non-contact force, the source mass and test mass interact with each other by means of a gravitational field. You can think of gravitational force as a “command” and gravitational field like the dialogue or speech used in giving the command.

Gravitational Field Definition

The space surrounding a material body in which gravitational force of attraction can be experienced is called its gravitational field.

What Is Gravitational Field Intensity?

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

E_g = F/m

Gravitational Field Intensity Due to a Point Mass (M) and test mass m is :

\(\begin{array}{l}E_g = \left[-\frac{\frac{GMm}{r^2}}{m}\right] \hat{r}\end{array} \)

Where G is the gravitational constant and r is the distance from the mass point mass M and test mass m.

⇒ Formula of Gravitational Field Intensity :

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

Where

\(\begin{array}{l}\hat{r}\ \text{represents the unit vector along the radial direction,}\end{array} \)

\(\begin{array}{l}\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\end{array} \)

\(\begin{array}{l}\hat{r}\ \text{represents the position vector of the test mass from the source mass.}\end{array} \)

The gravitational field intensity depends only upon the source mass and the distance of the unit test mass from 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 identical to the acceleration (preferably, we call it as acceleration due to gravity from the view of gravitation).

Points regarding Gravitational Field Intensity

• Defined as the force experienced by a unit mass (test mass) placed at any point in the gravitational field.
• The test mass itself is so small as compared to mass of planet (such as earth) that does not alter the gravitational field of planet.
• Given by the formula:
  • I = F_g/m or E_g = F_g/m
  • Where F is the gravitational force and m is the test mass.
• Numerically : g = E_g= GM/R²

Properties of Gravitational Field Intensity

1. Vector Quantity – Always directed towards the center of gravity.
2. Units – Newton per kilogram (N/kg) or meters per second squared (m/s²).
3. Dimensional Formula – [M⁰L¹T^-2].
4. Gravitational Field Intensity Due to a Point Mass (M) and test mass m is given by : \(\begin{array}{l}\mathbf{E_g = \left[-\frac{GM}{r^2}\right]\hat{r}}\end{array} \)
5. As Distance Increases, Intensity Decreases: E_g ∝ 1/r²
6. Intensity of gravitational field E_g = 0 , when r = ∞.
7. Zero Intensity Point:
   • If two bodies A and B have masses m₁ and m₂, at a certain distance apart, a point exists where the field intensities due to both cancel each other.
8. Gravitational Field Lines:
   • Represent the direction of the gravitational force.
   • For an isolated mass, they are radially inward.

Superposition Principle

The superposition principle extends to gravitational field intensities as,

E = E₁ + E₂ + E₃ + . . . . . . . + E_n

Where, E₁, E₂, E₃, . . . . . E_n are the gravitational field intensities at a point due to n particles in a system.

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

• Discrete mass distribution
• Continuous mass distribution

For a discrete mass distribution: 

\(\begin{array}{l}\vec{E} = \sum_{i = 1}^{n}E_{i}\end{array} \)

For a continuous mass distribution:

\(\begin{array}{l}\vec{E} = \int_{i}^{f} dE\end{array} \)

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

g = F/m

Where,

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

Gravitational Field Intensity of a Point Mass

Consider a point mass M, the gravitational intensity at a distance ‘r’ from it is given by

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

Frequently Asked Questions (FAQs)

Q1: What is the significance of the gravitational field intensity?

Ans: It helps determine the gravitational force acting on objects and affects planetary motion.

Q2: How does distance affect the intensity of the gravitational field?

Ans: The intensity decreases as the square of the distance from the mass increases.

Q3: What is the unit of gravitational field intensity?

Ans: Newton per kilogram (N/kg) or meters per second squared (m/s²).

Q4: Can the gravitational field intensity be zero?

Ans: Yes, at a point where the gravitational forces from two or more masses cancel each other.

Q5: What is the relation between gravitational field intensity and acceleration due to gravity?

Ans: They are numerically equal: g = E_g = GM/R²

Q6: What is a gravitational field?

The space or region around a body within which other bodies experience its gravitational force of attraction is called a gravitational field.

Q7: Define gravitational field intensity.

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

Solved Examples

Example 1: 

If the gravitational force and mass of a substance are 10 N and 5 kg, determine the gravitational field.

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 and m = 6 kg

The formula for gravitational field intensity is expressed by, g = F/m = 36/6 = 6 N/kg.

Multiple-Choice Questions (MCQs)

Q1: Which of the following statements is true about gravitational field intensity?

A) It is a scalar quantity
B) It increases with distance from the mass
C) It is always directed towards the center of mass
D) It has units of Joules

Correct Answer: C
Explanation: Gravitational field intensity is a vector quantity directed towards the center of mass.

Q2: The gravitational field intensity at a point due to a mass M at distance r is given by:

A) E_g = GM/R
B) E_g = GM/R²
C) E_g = GMR²
D) E_g = GMR

Correct Answer: B
Explanation: The formula for gravitational field intensity is E_g = GM/R²

Q3: At what point between two masses will the gravitational field intensity be zero?

A) Exactly at the midpoint
B) Closer to the larger mass
C) Closer to the smaller mass
D) At infinity

Correct Answer: C
Explanation: The zero-intensity point is closer to the smaller mass, as the larger mass exerts a stronger gravitational force.

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