Cambridge A-Level Physics 9702 – Gravitational Field
Gravitational Field
1. Definition of a Gravitational Field
A gravitational field is a region of space around a mass $M$ in which another mass $m$ experiences a force.
The field at any point is defined as the force per unit test mass placed at that point:
$$\mathbf{g} = \frac{\mathbf{F}}{m}$$
where $\mathbf{g}$ is the gravitational field strength (also called gravitational acceleration) and $\mathbf{F}$ is the gravitational force on the test mass.
2. Mathematical Form of the Gravitational Field of a Point Mass
For a point mass $M$, the field is radial and its magnitude follows Newton’s law of gravitation:
$$g = \frac{GM}{r^{2}}$$
where $G = 6.67\times10^{-11}\,\text{N m}^{2}\,\text{kg}^{-2}$ and $r$ is the distance from the centre of $M$.
The direction of $\mathbf{g}$ is always towards the source mass.
3. Representing a Gravitational Field with Field Lines
Field lines are a visual tool that help illustrate the direction and relative strength of a field.
The following rules apply to gravitational field lines:
Lines originate at infinity and terminate on the mass that creates the field (they never start or end in empty space).
All lines point towards the mass, indicating the attractive nature of gravity.
The density of lines (number per unit area) is proportional to the magnitude of the field; closer to the mass the lines are more closely spaced.
Field lines never cross; crossing would imply two different directions at the same point, which is impossible.
4. Sketching Field Lines for Common Configurations
Isolated point mass – Radial lines converge on the mass. The spacing decreases with decreasing $r$.
Two equal masses (binary system) – Lines emanate from each mass and meet midway, forming a saddle region where the net field is zero.
Earth’s surface – Near the surface the field can be approximated as uniform; parallel lines represent $g \approx 9.81\ \text{m s}^{-2}$.
Suggested diagram: Field lines around a single point mass and around two equal masses, showing line density and direction.
5. Quantitative Use of Field Lines
Although field lines are a qualitative tool, they can be linked to quantitative concepts:
The number of lines $N$ crossing a given surface $A$ is proportional to the flux $\Phi = \int \mathbf{g}\cdot d\mathbf{A}$.
For a spherical surface of radius $r$ centred on a point mass,
$$\Phi = g \, 4\pi r^{2} = \frac{GM}{r^{2}} \, 4\pi r^{2} = 4\pi GM,$$
which is constant, illustrating Gauss’s law for gravity.
6. Summary Table
Property
Gravitational Field
Field‑Line Representation
Source
Mass $M$ (point, sphere, planet)
Lines originate at infinity and terminate on $M$
Direction
Towards the source mass
Arrows on lines point inward
Magnitude
$g = GM/r^{2}$ (point mass)
Line density ∝ $g$ (closer → denser)
Superposition
Vector sum of individual fields
Resultant lines are the vector addition of contributions
Number of lines crossing any closed surface is constant
7. Common Misconceptions
Field lines are not physical objects; they are a representation.
The number of lines drawn is arbitrary; only relative density matters.
Even though lines converge on a mass, the field strength at the centre of a uniform sphere is zero (by symmetry).
8. Practice Questions
Draw the field‑line diagram for two masses $M_1 = 5\ \text{kg}$ and $M_2 = 10\ \text{kg}$ separated by $0.2\ \text{m}$. Indicate where the net field is zero.
Using Gauss’s law for gravity, calculate the gravitational flux through a spherical surface of radius $0.5\ \text{m}$ surrounding a $2\ \text{kg}$ point mass.
Explain how the field‑line density changes when moving from the surface of the Earth to an altitude of $400\ \text{km}$ (the approximate height of the International Space Station).
9. Further Reading
For deeper insight, consult the Cambridge International AS & A Level Physics (9702) syllabus sections on gravitation, and the textbook chapters covering Newton’s law of universal gravitation and Gauss’s law for gravity.