Field Strength g = GM/r²
Field Strength g = GM/r²
- The field strength puts a number on the field at each point: how many newtons per kilogram.
Definition and derivation
Symbols
- = gravitational field strength (N kg⁻¹ (= m s⁻²))
- = gravitational force on the test mass (N)
- = small test mass at the point (kg)
- = mass creating the field (kg)
- = distance from the centre of M (m)
- The one-mark definition: the gravitational field strength at a point is the force per unit mass on a small mass placed there (9702/41/O/N/25 Q3(a)).
- The two-mark derivation: put a test mass m at distance x, so ; divide by m to get , and say that G is the gravitational constant. Naming G is part of the mark (9702/41/O/N/25 Q3(b)).
- g is a vector, pointing at the mass creating the field. Near the Earth's surface it is the familiar 9.81 N kg⁻¹, the acceleration of free fall.
Worked example
Smallest case: the mass of the Earth
At the Earth's surface, g = 9.81 N kg⁻¹ and r = 6.4 × 10⁶ m. Find the Earth's mass (9702-style, from the coursebook).
- .
Answer
Scaling it, and adding two fields
- Inverse-square scaling: halve the distance and the field is four times stronger. A 2025 part asked exactly this, plus the direction: at the closer point the field is 4× as strong, and on the opposite side of the mass the two fields point opposite ways (9702/41/O/N/25 Q3(b)(iii)).
- Two masses: fields add as vectors. On the line between two equal spheres the total field is zero at the midpoint, and switches direction there: negative toward one mass, positive toward the other.
- The sketch marks for g between two equal spheres: starts at −g₀ at one surface, passes through zero at the midpoint, reaches +g₀ at the other, flattening near the middle and steepening near each sphere (9702/41/O/N/25 Q3(c)).
Common mistake
g is created by M and felt by the test mass. Do not confuse G (the universal constant, with units N m² kg⁻²) and g (the field strength at one place, N kg⁻¹).