Gravitational vs. Electrostatic Fields Revision Notes | A Level AQA Physics

1. Two Forces, One Mathematical Blueprint

Picture two entirely different scenarios: a planet pulling a satellite into orbit, and a proton attracting an electron inside an atom. The physical objects could not be more different — yet the equations governing both forces look almost identical. That structural similarity is not a coincidence, and understanding it is the heart of this topic.

Both Newton's law of gravitation and Coulomb's law are inverse-square laws. The force between two objects falls off with the square of the distance separating them. Double the separation, and the force drops to one quarter of its original value — in both cases.

Place these two equations side by side and the pattern is unmistakable: both have a product of source quantities in the numerator ($m_1 m_2$ or $Q_1 Q_2$), a constant of proportionality, and $r^2$ in the denominator. The mathematical skeleton is the same — only the physical quantities and constants differ.

2. Shared Descriptive Tools — Field Lines, Potential, and Equipotentials

Because the two force laws share the same mathematical structure, physicists use exactly the same set of descriptive tools for both fields. This is not just convenient — it reflects a genuine physical analogy between the two situations.

Consider field lines first. For a gravitational field, field lines point in the direction a small positive test mass would accelerate. For an electric field, field lines point in the direction a small positive test charge would accelerate. In both cases, the density of field lines represents the field strength: closer lines mean a stronger field.

The electric field has a direct analogue: electric field strength $E = F/Q$, the force per unit positive charge. Both $g$ and $E$ are defined using a test quantity (test mass or test charge) that is small enough not to disturb the field it is placed in. This parallel definition is deliberate — it means every technique you use to map one field transfers directly to the other.

The potential concept is powerful because it is a scalar — you can add potentials from multiple sources by ordinary arithmetic, without worrying about vector directions. This works identically in both gravitational and electric contexts.

Equipotential surfaces complete the toolkit. These are surfaces on which the potential is constant, so no work is done moving a test mass or test charge along them. In both fields, equipotential surfaces are always perpendicular to field lines. For a point mass or point charge, the equipotentials are concentric spheres — the same geometry in both cases.

3. The Critical Difference — Attraction Only vs Attraction and Repulsion

Here is where the analogy breaks down in one fundamental way. Mass only ever comes in one sign — there is no such thing as negative mass. This means gravitational forces are always attractive: every mass attracts every other mass, without exception.

Charge, by contrast, comes in two signs: positive and negative. When two charges have the same sign (both positive or both negative), the electrostatic force between them is repulsive. When they have opposite signs, the force is attractive. The sign of the product $Q_1 Q_2$ in Coulomb's law encodes this: a positive product means repulsion, a negative product means attraction.

This single difference has enormous physical consequences. Gravitational fields cannot be shielded or cancelled — you cannot place a negative mass between two positive masses to neutralise the attraction. Electric fields, however, can be screened: a conductor redistributes its charges to cancel the internal field, and opposite charges can neutralise each other entirely.

There is a second consequence worth noting at A-level. Because gravity is always attractive and there is no cancellation, gravitational fields dominate at astronomical scales — the cumulative pull of enormous masses is never offset. Electrostatic forces, despite being vastly stronger at the atomic scale, tend to cancel out over large distances because matter is overall electrically neutral.

4. Comparing Field Line Diagrams — What the Geometry Reveals

Field line diagrams for the two field types look similar in some configurations and strikingly different in others. For a single point mass, the gravitational field lines point radially inward — there is no outward option, because gravity only attracts. For a single positive point charge, the electric field lines point radially outward; for a negative charge, they point radially inward.

The diagram for a negative point charge and the diagram for a point mass are therefore geometrically identical: radial lines converging on the source. This is not coincidental — it reflects the mathematical equivalence of $V_g = -Gm/r$ and the electric potential for a negative charge. If the charge is $Q = -|Q|$, then $V = kQ/r = k(-|Q|)/r = -k|Q|/r$, which is also negative. The potential in both cases is negative and increases (becomes less negative) with distance.

Where the diagrams diverge is when you introduce a second source. Two positive charges produce a field with a neutral point between them where the fields cancel — a configuration impossible in gravitation, because two masses always reinforce each other's pull rather than opposing it.

5. Worked Comparison — Applying the Analogy

The real test of understanding this analogy is being able to transfer reasoning from one field to the other. The following worked example shows how a question about one field type can be answered by drawing on knowledge of the other — a skill examiners reward explicitly.

Worked example: Comparing Force Ratios in Gravitational and Electrostatic Fields

A student doubles the separation between two point masses from $r$ to $2r$. A second student doubles the separation between two point charges of the same sign from $r$ to $2r$. Compare the effect on the magnitude of the force in each case, and explain why the same reasoning applies to both.

1. Gravitational: $F_g = \frac{Gm_1 m_2}{r^2}$ Electrostatic: $F_e = \frac{kQ_1 Q_2}{r^2}$ — Both forces share the same $1/r^2$ dependence. The source quantities and constants differ, but the $r^2$ structure is identical.
2. Gravitational: $F_g' = \frac{Gm_1 m_2}{(2r)^2} = \frac{Gm_1 m_2}{4r^2}$ Electrostatic: $F_e' = \frac{kQ_1 Q_2}{(2r)^2} = \frac{kQ_1 Q_2}{4r^2}$ — $(2r)^2 = 2r \times 2r = 4r^2$. The denominator increases by a factor of 4 in both cases.
3. $\frac{F'}{F} = \frac{1/4r^2}{1/r^2} = \frac{1}{4}$ — The force is reduced to one quarter of its original value in both the gravitational and electrostatic cases.
4. In both cases: doubling $r$ reduces $F$ by a factor of 4. — The same reasoning applies because both laws are inverse-square laws with $r^2$ in the denominator. The physical quantities ($m$ vs $Q$) and constants ($G$ vs $k$) are different, but the mathematical structure is identical, so the proportional effect of changing $r$ is the same.

Answer: Doubling the separation reduces the magnitude of the force to one quarter ($\times \frac{1}{4}$) in both the gravitational and electrostatic cases, because both obey an inverse-square law: $F \propto 1/r^2$.

6. Exam Technique — Comparing and Evaluating the Two Fields

Questions on this topic frequently use the command words compare, explain, and evaluate. Each demands a different response structure, and confusing them is one of the most common ways students lose marks.

A compare question requires you to address both similarities and differences explicitly. Stating only similarities, or only differences, will not earn full marks. Structure your answer as: similarity — difference — similarity — difference, or use a clear two-part structure with a heading for each.

An explain question requires a causal chain: observation + 'because' + scientific principle. For example: 'Gravitational field lines always point towards the source mass because mass is always positive and gravity is always attractive.' The word 'because' is doing the work that earns the mark.

The core message of this topic is that the inverse-square law is a unifying mathematical principle: it generates the same descriptive toolkit — field lines, potential, and equipotential surfaces — regardless of whether the source is mass or charge. The one place the analogy fails is physical, not mathematical: gravity has no repulsive mode, because mass carries no sign.