Gravitational Field Strength

Every object with mass produces a gravitational field around itself — a region of space in which any other mass will experience a gravitational force. The field is invisible, but its effects are familiar: a dropped book falls to the floor, the Moon orbits the Earth, the Earth orbits the Sun. All of these are the action of gravitational fields.

This lesson introduces the concept of the gravitational field and the quantity that measures its strength: g. It is the starting point of OCR A-Level Physics A Module 5.4 (Gravitational fields).

---

The Field Concept

In 19th-century physics, forces between distant objects were thought of as "action at a distance" — mysterious tugs across empty space. Michael Faraday, thinking about electromagnetism, proposed a different picture: each mass (or charge) fills the space around it with a field, a vector quantity defined at every point, and another mass (or charge) placed in the field feels a force because of the field at its location, not because of a direct connection to the distant source.

This field picture is now standard throughout physics. For gravity, the field at a point tells us how a test mass placed there would be pulled.

---

Definition of Gravitational Field Strength

The gravitational field strength at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point:

g = F / m

Symbol: g. SI unit: newtons per kilogram (N kg⁻¹), which is exactly equivalent to metres per second squared (m s⁻²), since F/m = ma/m = a.

Gravitational field strength is a vector: it has the same direction as the force on the test mass, i.e. towards the source.

Exam Tip: OCR loves to ask, "State the unit of gravitational field strength." Either N kg⁻¹ or m s⁻² is acceptable — they are dimensionally identical. The N kg⁻¹ form is preferred because it emphasises that g is a force per unit mass.

Does the Test Mass Matter?

For an everyday mass on Earth, the force of gravity is proportional to its mass, so F/m comes out the same for every test mass — the field is a property of the source, not the test object. (Strictly, a very large test mass would perturb the source, but we ignore that by taking "test masses" to be small.)

---

Field Lines

A convenient way to visualise a gravitational field is with field lines — directed curves that show the direction of the gravitational force on a test mass at each point. Two rules:

1. The field is along the field line at each point, in the direction of the arrow.
2. The density of field lines (lines per unit area perpendicular to the field) is proportional to the field strength.

Where the lines are close together, the field is strong; where they are far apart, the field is weak.

Uniform Fields

Near the Earth's surface — on length scales much smaller than the Earth's radius — the gravitational field is almost perfectly uniform. The field strength is essentially constant at g = 9.81 N kg⁻¹ and the direction is vertically downward.

graph TD
    U1[↓] --- U2[↓] --- U3[↓] --- U4[↓]
    U5[↓] --- U6[↓] --- U7[↓] --- U8[↓]
    U9[↓] --- U10[↓] --- U11[↓] --- U12[↓]
    G[Earth surface]

Field lines are parallel, equally spaced, and pointing straight down. That is why mgh is a reliable formula for gravitational PE only when the height h is small compared with the Earth's radius.

Radial Fields

On a larger scale, the gravitational field of a spherical mass such as a planet or star is radial — the field lines all point inward, converging on the centre.

graph TD
    A[↘] --- O((Planet)) --- B[↙]
    C[↓] --- O
    D[↖] --- O --- E[↗]
    F[←] --- O --- G[→]
    H[↑] --- O

For a radial field, the field lines get closer together as you approach the planet — and the field strength grows correspondingly stronger. Far from the planet, the lines are widely spread and the field is weak.

---

g Near the Earth's Surface

Everyone knows that g ≈ 9.81 N kg⁻¹ near the Earth's surface. OCR will sometimes use 9.8 or 10 to simplify arithmetic. What is less obvious is that this number depends slightly on where you are: