You already know from static electricity that charged objects push and pull on one another without touching — a comb lifts scraps of paper, like charges shove apart, unlike charges snap together. But how strong is that force, exactly? How does it fade as the charges move apart? And what invisible thing reaches across the empty gap to carry the push?

At A-level we answer all three the same way Newton answered them for gravity. Charge, like mass, fills the space around it with a field — a region where any other charge feels a force — and that field weakens with distance in exactly the same inverse-square way. The parallel is so close that once you have the gravity picture in your bones, electric fields fall almost straight out of it — with one dramatic new twist: charges come in two signs, so electric forces can repel as well as attract.

Coulomb's law: the force between two charges

In the 1780s Charles-Augustin de Coulomb measured the force between two small charged spheres with a delicate torsion balance and found a law that looks uncannily like Newton's law of gravitation. The force grows with each charge and shrinks with the square of their separation.

Unlike G, the electric constant is enormous — about 8.99\times10^{9} versus gravity's 6.67\times10^{-11}. That is why the electric force between two protons is some 10^{36} times stronger than their gravitational pull, and why it is electricity, not gravity, that holds atoms and molecules — and you — together.

Worked example — the force between two charges

Two small spheres carry Q_1 = +3\ \mu\text{C} and Q_2 = +2\ \mu\text{C}, held r = 0.50\ \text{m} apart. Find the force between them.

Step 1 — write Coulomb's law.

Step 2 — substitute (charges in coulombs: 3\ \mu\text{C} = 3\times10^{-6}\ \text{C}).

Step 3 — evaluate. The top is (8.99\times10^{9})(6\times10^{-12}) = 5.39\times10^{-2}, and the bottom is 0.25, so

Both charges are positive — like — so this force is repulsive: it pushes the two spheres apart. Had one charge been negative, the same size of force would pull them together. Gravity never gets that choice: mass only ever attracts.

The electric field of a point charge

Rather than recompute the force for every different test charge, we describe the space itself once. The electric field strength E at a point is the force a unit positive charge would feel there — a property of the location, set by the source charge, not by whatever we drop into it:

For a single point (or spherical) charge Q, putting Coulomb's law over one coulomb of test charge gives the radial field

which — just like gravity's g = GM/r^2 — is a vector that obeys the inverse-square law and depends only on the source charge and the distance from it. Its direction is where the sign steps in: because E is the force on a positive charge, the field points away from a positive source (a positive test charge is repelled outward) and towards a negative source (a positive test charge is pulled inward).

Worked example — field near a point charge

A point charge Q = +5\ \mu\text{C} sits in a vacuum. Find the field strength a distance r = 0.10\ \text{m} away.

The field points radially outward (the source is positive). Drop a charge q = -2\ \text{nC} at that spot and it feels a force F = Eq = (4.5\times10^{6})(2\times10^{-9}) \approx 9.0\times10^{-3}\ \text{N}, directed towards the source because the test charge is negative.

Field lines: picturing the field

We draw a field as a set of field lines — arrows showing the direction a positive test charge would be pushed. For a point charge the lines are radial: straight spokes running out of a positive charge and into a negative one. Where the lines crowd together the field is strong; where they spread apart it is weak — which is exactly why the field is fiercest close in and fades with distance. Flip the sign below and watch every arrow reverse.

Two rules never break: field lines start on positive charge and end on negative charge (or run off to infinity), and they never cross — if they did, the field would point two ways at once at the crossing point, which is nonsense.

See the inverse-square fall

The curve below is E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2} for a +1\ \text{nC} charge (so \tfrac{1}{4\pi\varepsilon_0}Q \approx 9 and E \approx 9/r^2\ \text{N/C}). Drag the slider outward and watch how fast the field collapses: it is already quartered by r = 2\ \text{m} and down to a ninth by r = 3\ \text{m}. That steep, never-quite-zero shape is the fingerprint of every inverse-square law — gravity and electricity share it exactly.

The uniform field between parallel plates

Bring two flat metal plates close together and connect them to a supply, so one is at a higher potential than the other. Between them the field is completely different from the radial spray of a point charge: the field lines are straight, parallel and evenly spaced, all running from the positive plate to the negative one. Evenly spaced lines mean a uniform field — the same strength everywhere in the gap — given by the beautifully simple

where V is the potential difference between the plates and d their separation. This is where the alternative unit V/m for field strength comes from. Slide the voltage and separation below and watch the field readout respond.

Worked example — deflecting a charge

Two plates d = 0.05\ \text{m} apart have a potential difference of V = 200\ \text{V}. Find the field, and the force on an electron (q = 1.6\times10^{-19}\ \text{C}) passing between them.

Tiny — but an electron is so light that this steady sideways push flings it into a smooth parabolic curve, exactly like a ball thrown in gravity's uniform field near the ground. That is the trick behind the old cathode-ray tube and the ink-jet printer.

The gravity–electricity analogy

Line the two field laws up side by side and the resemblance is almost complete. Learn one column and you very nearly get the other for free — the algebra is identical; only the source quantity, the constants, and (crucially) the possibility of repulsion change.

The single deep difference sits in the last row. Mass comes in only one flavour, so gravity always pulls inward. Charge comes in two, so an electric field can drive a charge either way — and that is the whole reason electricity can do such intricate work that gravity never could.