Gravitational Fields, Potential, and Equipotentials

The gravitational potential V at a point is the work done per unit mass in bringing a small mass from infinity to that point, V = −GM ÷ r for a spherical mass M; the field strength is the negative gradient of the potential, g = −dV/dr, and equipotential surfaces (surfaces of constant V) are everywhere perpendicular to the field lines. In brief: the potential is negative everywhere and rises toward zero at infinity. The gravitational potential energy of a mass m is U = mV. The work needed to move a mass between two points is W = mΔV, independent of the path (gravity is a conservative force). Closer equipotentials mean a stronger field. The potentials of several masses add as scalars, which is simpler than adding fields as vectors.

This guide gives a more advanced treatment of the gravitational field than the introductory one: it defines the potential and its relation to the field, shows the shape of equipotentials and field lines for single and double masses, explains potential graphs and the area under a g-r graph, compares field strength and potential in a table, works through numerical examples and corrects common mistakes.

What is gravitational potential?

Moving a mass away from a planet requires work against gravity. The gravitational potential V at a point is defined as the work done per unit mass by an external agent in moving a test mass from infinity to that point, so that the zero is at infinity. Since the force is attractive, the work done by the agent is negative, and V is negative:

V = −GM ÷ r, in J/kg.

At the surface of the Earth, V = −3.986 × 10¹⁴ ÷ 6.371 × 10⁶ = −6.26 × 10⁷ J/kg (−62.6 MJ/kg). The potential energy of a mass m is U = mV. The more negative the potential, the deeper “in” the gravitational well. For several masses, the total potential is the sum of the individual potentials (a scalar sum): V = −G(M₁/r₁ + M₂/r₂ + …). Between the Earth and Moon, the potential reaches a maximum at the point where the fields cancel (about 90 percent of the way to the Moon), the “saddle point” that a spacecraft must cross.

How are potential and field related?

The work done by gravity in moving a mass m through a small distance dr is F dr = −m dV, so the field strength is the negative of the potential gradient: g = −dV/dr. For V = −GM/r, differentiating gives g = −GM/r² (the minus sign shows that the field points toward the center, in the direction of decreasing r). Conversely, the change in potential between two points is the area under the g-r graph: ΔV = −∫g dr. In a uniform field near the surface of a planet, ΔV = gΔh, which gives ΔU = mgΔh. The gradient relation means that the field points “downhill” in the potential, in the direction in which the potential decreases most rapidly, as a ball rolls downhill on a landscape: the potential is like the height of the ground in a relief map, and the field like the slope.

What are equipotentials?

An equipotential surface connects points with the same gravitational potential. For an isolated spherical mass, the equipotentials are concentric spheres. No work is done in moving a mass along an equipotential surface, which means that the field has no component along the surface: the field lines cross the equipotentials at right angles. In a diagram, equipotentials are drawn at equal intervals of potential, as with contour lines on a map: where they are close together, the field is strong. Near the surface of a planet, over a small region, the equipotentials are nearly parallel horizontal planes, equally spaced for equal height intervals (uniform field). For two masses, the equipotentials around each are close to circles, then merge into a figure-of-eight shape and surround both masses at larger distances. Mean sea level on Earth is approximately an equipotential surface (the geoid), slightly flattened by the Earth’s rotation and irregular because of the uneven distribution of mass.

What do the graphs look like?

• g against r (outside a sphere): a curve that falls as 1/r². Inside a uniform sphere, g rises linearly from zero at the center to the maximum at the surface. The area under the curve gives the potential difference.
• V against r: a curve with V = −GM/r that rises from a large negative value at the surface and approaches zero at infinity. Its gradient is −g (positive: V increases with r). Inside a uniform sphere, the potential varies as a parabola, with the value −3GM/2R at the center.

Field strength and potential compared

Worked example: potential differences and work

1. Potential at the surface and in orbit. GM_E = 3.986 × 10¹⁴ m³/s². At the surface (6.371 × 10⁶ m): V = −62.6 MJ/kg. At 400 km altitude (6.771 × 10⁶ m): V = −58.9 MJ/kg. The difference: ΔV = +3.70 MJ/kg.
2. Work to lift a satellite. For a satellite of mass 1,000 kg: ΔU = mΔV = 1,000 × 3.70 × 10⁶ = 3.7 × 10⁹ J of potential energy to be supplied (not counting the kinetic energy of the orbit, which is 2.9 × 10¹⁰ J). The estimate mgΔh = 1,000 × 9.8 × 4.0 × 10⁵ = 3.9 × 10⁹ J is 6 percent too high, since g decreases with height.
3. Field from the potential. Between r = 6.371 × 10⁶ and 6.771 × 10⁶ m, the average gradient is ΔV/Δr = 3.70 × 10⁶ ÷ 4.0 × 10⁵ = 9.25 N/kg, which is the mean field in this range (9.82 at the bottom and 8.69 at the top).
4. Potential of two bodies. At the surface of the Earth, the potential due to the Moon is −GM_M/d = −4.905 × 10¹² ÷ 3.84 × 10⁸ = −1.3 × 10⁴ J/kg, about 0.02 percent of the Earth’s own contribution, so the total is −62.6 MJ/kg − 0.013 MJ/kg.
5. Potential energy of a spacecraft. A 500 kg probe at a distance of 3.0 × 10⁷ m from the center of the Earth: V = −3.986 × 10¹⁴ ÷ 3.0 × 10⁷ = −1.33 × 10⁷ J/kg, so U = −6.6 × 10⁹ J. The energy to remove it to infinity: +6.6 × 10⁹ J (plus kinetic energy).
6. Moving along an equipotential. A satellite in a circular orbit stays on a sphere of constant r, an equipotential, so gravity does no work on it, and its kinetic energy stays constant.

How were these concepts developed?

The concept of potential was introduced by Joseph-Louis Lagrange (1773) and Pierre-Simon Laplace, who showed that the potential satisfies Laplace’s equation in empty space; Siméon Poisson extended it to regions with mass (1813), and George Green and Carl Friedrich Gauss developed the theory in the 1820s to 1840s. The geoid is measured with satellites: GRACE (2002 to 2017) and GOCE (2009 to 2013) mapped the Earth’s gravitational field in detail. In general relativity, the gravitational potential appears in the metric of spacetime.

Which mistakes are common?

Mistake 1: Treating the potential as positive

With the zero at infinity, V is negative for any distance. The sign shows that the system is bound.

Mistake 2: Adding the potentials as vectors

V is a scalar. Add values with their signs. Fields are vectors.

Mistake 3: Confusing potential with potential energy

Potential is energy per unit mass (J/kg). Potential energy is U = mV.

Mistake 4: Thinking that a zero field means zero potential

At the null point between two masses, g = 0 but V is not zero: it is at a maximum along the line joining them.