represent a gravitational field by means of field lines

Gravitational Field – Cambridge International AS & A Level Physics (9702)

1. Definition and Vector Nature (AO1)

A gravitational field is the region of space surrounding a mass \(M\) in which another mass \(m\) experiences a gravitational force.

The field at any point is defined as the force per unit test mass placed at that point:
\[
\mathbf{g}=\frac{\mathbf{F}}{m}
\]

According to the syllabus, gravitational field strength (\(\mathbf{g}\)) is a vector quantity. Its magnitude is \(g=|\mathbf{g}|\) and its direction is the direction of the force on a positive test mass (i.e. towards the source mass).

2. Derivation from Newton’s Law of Gravitation (AO2)

Newton’s law for the force between two point masses is
\[
\mathbf{F}= -\,\frac{G M m}{r^{2}}\;\hat{\mathbf r}
\]
where \(\hat{\mathbf r}\) is the outward radial unit vector and the minus sign shows the force is attractive.

Dividing by the test mass \(m\) gives the field produced by the source mass \(M\):
\[
\boxed{\;\mathbf{g}= -\,\frac{G M}{r^{2}}\;\hat{\mathbf r}\;}
\]
Hence the magnitude is \(g=\dfrac{G M}{r^{2}}\) and the direction is radially inward.

Constant: \(G = 6.67\times10^{-11}\;\text{N·m}^{2}\text{kg}^{-2}\).

3. Gravitational Potential and Its Relation to the Field (AO2)

The gravitational potential \(\Phi\) (sometimes written \(V\)) at a distance \(r\) from a point mass is
\[
\Phi(r)= -\,\frac{G M}{r}
\]

The field is the negative gradient of the potential:
\[
\mathbf{g}= -\nabla\Phi
\]
For a spherically symmetric source this reduces to the expression in section 2.

4. Gravitational Field Inside a Uniform Solid Sphere (AO1)

Inside a uniform solid sphere of total mass \(M\) and radius \(R\) the field varies linearly with distance \(r\) from the centre:
\[
\mathbf{g}(r)= -\,\frac{G M}{R^{3}}\,r\;\hat{\mathbf r}\qquad (0\le r\le R)
\]

Consequences:
- At the centre (\(r=0\)) the field is zero.
- At the surface (\(r=R\)) the expression reduces to the point‑mass result \(g=GM/R^{2}\).

5. Representing a Gravitational Field with Field Lines (AO1)

Field lines are a visual aid that convey the direction and *relative* strength of a vector field.

Rule	Meaning for a Gravitational Field
Origin and termination	Lines start at infinity and terminate on the mass that creates the field; they never begin or end in empty space.
Direction	Arrows on the lines point inward, reflecting the attractive nature of gravity.
Density	The density of lines (number per unit area) is qualitatively proportional to the magnitude of \(\mathbf{g}\). Closer to the source the lines are more densely packed.
Crossing	Lines never cross; a crossing would imply two different directions at the same point, which is impossible for a vector field.
Superposition	For several masses the resultant field is obtained by vector addition of the individual fields; the combined line pattern reflects this addition.

6. Sketches of Field‑Line Diagrams for Common Configurations (AO1)

Isolated point mass – Radial lines converge on the mass. The spacing decreases as \(r\) decreases, illustrating the \(1/r^{2}\) dependence.

Two equal masses (binary system) – Each mass emits its own set of lines. Mid‑way a neutral (saddle) point occurs where the fields cancel; the lines are equally spaced on either side of this point.

Two unequal masses – The neutral point is nearer to the smaller mass. The denser set of lines belongs to the larger mass.

Earth’s surface (or any small region of a large sphere) – Over a limited area the field may be approximated as uniform; parallel lines represent \(g\approx9.81\;\text{m s}^{-2}\).

Uniform spherical shell – No field lines enter the interior; the field inside the shell is zero.

Uniform solid sphere (inside) – Lines emerge from the centre with a spacing that increases linearly with radius, reflecting the linear increase of \(g\) with \(r\) (section 4).

Diagram placeholders (to be drawn by the student):

Figure 1 – Radial lines for a single point mass.

Figure 2 – Lines for two equal masses showing the neutral point.

Figure 3 – Lines for two unequal masses with the shifted neutral point.

Figure 4 – Parallel lines near Earth’s surface.

Figure 5 – No lines inside a spherical shell.

Figure 6 – Linear‑density lines inside a uniform solid sphere.

7. Gravitational Gauss’s Law (AO2)

The total gravitational flux through any closed surface that encloses a point mass \(M\) is
\[
\boxed{\;\displaystyle\oint \mathbf{g}\!\cdot\!d\mathbf{A}=4\pi G M\;}
\]
This result is independent of the shape or size of the surface.

For a spherical surface of radius \(r\):
\[
\Phi = g\,(4\pi r^{2}) = \frac{G M}{r^{2}}\,(4\pi r^{2}) = 4\pi G M .
\]

In the syllabus the emphasis is on the *qualitative* link: a greater number of field lines crossing a surface indicates a larger flux; the exact “flux per line” is not required.

8. Superposition of Gravitational Fields (AO1 & AO2)

For several masses the total field at any point is the vector sum:
\[
\mathbf{g}{\text{total}} = \sum{i}\mathbf{g}_{i}
\]

When drawing field lines for multiple sources, first sketch the individual patterns, then adjust the lines so that they follow the resultant direction at each point.

9. Summary Table (Quick Revision)

Aspect	Mathematical description	Field‑line representation
Source	Mass \(M\) (point, sphere, shell)	Lines terminate on \(M\); start at infinity.
Direction	\(\mathbf{g}\) points toward the source.	Arrows on lines point inward.
Magnitude (point mass)	\(g = \dfrac{G M}{r^{2}}\)	Line density ∝ \(g\); denser nearer the mass.
Inside uniform solid sphere	\(\mathbf{g}(r)= -\dfrac{G M}{R^{3}}\,r\;\hat{\mathbf r}\)	Lines emerge from centre with spacing increasing linearly with \(r\).
Potential	\(\Phi = -\dfrac{G M}{r}\)	Equipotential surfaces are concentric spheres; field lines are perpendicular to them.
Superposition	\(\mathbf{g}{\text{total}} = \sumi \mathbf{g}_i\)	Resultant pattern is the vector sum of the individual line sets.
Gauss’s law (flux)	\(\displaystyle\oint \mathbf{g}\!\cdot\!d\mathbf{A}=4\pi G M\)	Same total number of lines cross any closed surface surrounding the mass.

10. Common Misconceptions (AO1)

Field lines are physical objects. They are a pictorial aid; the gravitational field exists everywhere, whether or not we draw lines.

Number of drawn lines matters. Only the *relative* density conveys information about field strength; the absolute number is arbitrary.

Field inside a uniform solid sphere. Many think the field is strongest at the centre; in fact it is zero at the centre and increases linearly with radius.

Neutral point in a binary system. It lies closer to the smaller mass, not exactly midway.

Equipotential surfaces are the same as field lines. Equipotentials are perpendicular to field lines; they are not the same thing.

11. Practice Questions (AO1 – AO3)

Diagramming (AO1) – Draw the field‑line diagram for two masses \(M{1}=5\;\text{kg}\) and \(M{2}=10\;\text{kg}\) separated by \(0.20\;\text{m}\). Mark the point where the net field is zero and explain why it is nearer to the smaller mass.

Gauss’s law calculation (AO2) – Using gravitational Gauss’s law, find the flux through a spherical surface of radius \(0.50\;\text{m}\) that encloses a point mass of \(2\;\text{kg}\). (Answer: \(\Phi = 4\pi G(2\;\text{kg})\)).

Field‑line density with altitude (AO1) – Explain qualitatively how the spacing of the lines changes when moving from the Earth’s surface to an altitude of \(400\;\text{km}\) (ISS height). Use the \(1/r^{2}\) dependence to justify the change.

Superposition problem (AO2) – Two equal masses are placed \(0.10\;\text{m}\) apart. Determine the magnitude and direction of the gravitational field at a point midway between them using vector addition, then illustrate the result with field lines.

Potential and field relation (AO2) – A point mass \(M=3\;\text{kg}\) is at the origin. Write the expression for the gravitational potential \(\Phi(r)\) and show, by differentiating, that \(\mathbf{g}= -\nabla\Phi\) yields the field formula of section 2.

Orbital motion application (AO3) – An artificial satellite moves in a circular orbit of radius \(r=7.0\times10^{6}\;\text{m}\) around Earth (mass \(M_{\earth}=5.97\times10^{24}\;\text{kg}\)). Using \(g = v^{2}/r\) and the expression for \(\mathbf{g}\), calculate the orbital speed \(v\). Comment on how the field‑line picture (parallel lines near the surface) relates to the uniform‑field approximation used in low‑orbit calculations.

12. Further Reading (AO1)

Cambridge International AS & A Level Physics (9702) – Syllabus sections 5.1–5.4 (Gravitation).

Textbook: Physics for Cambridge International A Level, chapters on Newton’s law of gravitation, gravitational potential, and Gauss’s law for gravity.

Online:
- Cambridge Assessment International Education – “Gravitational fields and potentials” tutorial videos.
- Khan Academy – “Gravitational field and potential” playlists (useful for visualising field lines).

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