represent a gravitational field by means of field lines

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

1. Isolated point mass – Radial lines converge on the mass. The spacing decreases with decreasing \$r\$.
2. Two equal masses (binary system) – Lines emanate from each mass and meet midway, forming a saddle region where the net field is zero.
3. Earth’s surface – Near the surface the field can be approximated as uniform; parallel lines represent \$g \approx 9.81\ \text{m s}^{-2}\$.

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

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

1. Draw the field‑line diagram for two masses \$M1 = 5\ \text{kg}\$ and \$M2 = 10\ \text{kg}\$ separated by \$0.2\ \text{m}\$. Indicate where the net field is zero.
2. 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.
3. 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.