Gravitational Fields

Gravity is the force of attraction between any two objects with mass. Although it is the weakest of the four fundamental forces, it dominates on the largest scales because it acts over infinite range and is always attractive. Unlike the electric force, gravity cannot be shielded or repulsive. This topic is tested in AQA Paper 2 (Section 7) and OCR Paper 1 (Module 5), and draws many parallels with electric fields — examiners frequently test your ability to compare the two.

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Newton's Law of Universal Gravitation

Key Definition: Newton's Law of Universal Gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

F = −GMm/r²

where G is the gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²), M and m are the two masses, and r is the distance between their centres. The negative sign indicates the force is always attractive (directed towards the other mass).

This is an inverse square law — doubling the distance reduces the force to one quarter; tripling the distance reduces it to one ninth.

Important points:

• The law applies to point masses or to uniform spherical masses (which behave as though all their mass is concentrated at their centre).
• The gravitational force between everyday objects (e.g. two people) is negligibly small because G is such a tiny number.
• The force is mutual — the Earth pulls on you with the same magnitude of force as you pull on the Earth (Newton's Third Law).

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Gravitational Field Strength

Key Definition: The gravitational field strength (g) at a point is the gravitational force per unit mass experienced by a small test mass placed at that point.

g = F/m = GM/r²

The units of g are N kg⁻¹, which are dimensionally equivalent to m s⁻² (the acceleration due to gravity). Near the Earth's surface, g ≈ 9.81 N kg⁻¹. As you move away from the Earth, g decreases with the square of the distance from the centre of the Earth.

Gravitational Field Line Diagrams

Radial field (around a spherical mass such as a planet or star):

• Field lines point radially inward towards the centre of the mass.
• The lines are straight and converge at the centre.
• The lines are more widely spaced further from the mass, indicating that the field strength decreases with distance.
• The field has spherical symmetry — it looks the same from any direction.

Described diagram — Radial gravitational field lines around a planet: A circle represents the planet at the centre. Straight arrows point radially inward from all directions towards the centre of the planet, evenly distributed around the circumference. Near the surface, the arrows are closely spaced, indicating a strong field. Further from the planet, the arrows are more widely spaced, showing the field weakening with distance. Concentric dashed circles may be drawn around the planet to represent equipotential surfaces, perpendicular to the field lines at every point.

Uniform field (near the Earth's surface):

• Field lines are parallel, equally spaced, and point vertically downwards.
• The spacing is constant, indicating that g is approximately constant over small height differences.
• This is an approximation valid when the height change is very small compared to the radius of the Earth.

Exam Tip: When drawing radial field lines, always use arrows pointing towards the centre of the mass. The lines should never cross. For a uniform field, the lines must be parallel and evenly spaced — uneven spacing implies a non-uniform field.

The g-r Graph

A graph of gravitational field strength (g) against distance from the centre of a uniform sphere (r) has two distinct regions:

• Inside the sphere (r < R): g increases linearly from zero at the centre to a maximum at the surface. This is because only the mass enclosed within radius r contributes to the field at that point.
• Outside the sphere (r > R): g decreases as 1/r² (an inverse square relationship). The curve is a smooth hyperbola-like shape that tends towards zero as r tends to infinity.

At the surface (r = R), g has its maximum value: g = GM/R².

Described diagram — g-r graph (gravitational field strength against distance): The horizontal axis is distance from the centre of the sphere (r) and the vertical axis is gravitational field strength (g). For r < R (inside the sphere), the graph is a straight line rising from the origin to a maximum at r = R. For r > R (outside the sphere), the curve decreases steeply following an inverse square relationship (g ∝ 1/r²), creating a smooth curve that tends towards zero as r becomes very large. There is a clear peak at r = R, and the transition from the linear region to the 1/r² curve is smooth at the surface.

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Gravitational Potential

Key Definition: The gravitational potential (V) at a point is the work done per unit mass in bringing a small test mass from infinity to that point.

V = −GM/r

Gravitational potential is always negative because:

• We define V = 0 at infinity.
• The gravitational force is attractive, so the test mass naturally falls inward — work is done by the field, not against it.
• To remove the mass back to infinity, positive work must be done (energy input required).

The most negative potential is at the surface of the mass (closest approach), and V increases (becomes less negative) as r increases, approaching zero at infinity.

The V-r Graph

A graph of gravitational potential (V) against distance from the centre (r) for a point or spherical mass:

• The curve starts at a large negative value near the surface and increases (becomes less negative) as r increases.
• The curve is asymptotic to V = 0 as r approaches infinity.
• The curve has a 1/r shape (not 1/r²).
• Inside a uniform solid sphere, V continues to become more negative towards the centre, reaching its most negative value at r = 0.

Described diagram — V-r graph (gravitational potential against distance): The horizontal axis is distance from the centre (r) and the vertical axis is gravitational potential (V), with all values below the horizontal axis because V is always negative. Near the surface, V has a large negative value. As r increases, the curve rises smoothly towards zero but never reaches it — it is asymptotic to V = 0. The curve has a 1/r shape (less steep than the 1/r² shape of the g-r graph). The most negative point on the curve corresponds to the surface of the mass. A tangent drawn to the curve at any point has a gradient equal to −g at that distance.

Equipotential Surfaces

• Equipotential surfaces connect points of equal gravitational potential.
• For a spherical mass, they are concentric spheres.
• Equipotential surfaces are always perpendicular to field lines.
• No work is done when a mass moves along an equipotential surface.
• The surfaces are closer together where the field is stronger (near the mass) and further apart where the field is weaker.

Potential Difference and Energy

The gravitational potential energy of a mass m at distance r from a mass M is:

Eₚ = mV = −GMm/r

The change in potential energy when moving between two points is:

ΔEₚ = mΔV

This equation is extremely useful for calculating the energy needed to move a satellite from one orbit to another.

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Relationship Between g and Potential Gradient