Coulomb's Law

It is worth pulling the law apart into its three separate experimental claims, because each is tested differently and each fails differently. Claim one is geometric: the force on a charge held at rest lies along the straight line to the other charge, with no sideways component at all, so the direction of every term in a problem is fixed before any arithmetic begins. Claim two is linearity in charge: multiply either charge by a factor and the force scales by that same factor, which is why only the product |q₁||q₂| ever appears. Claim three is the exponent: the force falls as 1/r², with r measured centre to centre and not between the surfaces. Coulomb himself got the exponent from a torsion balance in 1785, reading the twist of a fine wire as he moved two charged pith balls. Modern tests never measure a force at all — they hunt for an electric field inside a closed charged conductor, which is exactly zero only if the exponent is exactly 2, and they bound any departure from 2 to about one part in 10¹⁶.

Write k = 1/(4πε₀) = 8.988 × 10⁹ N m² C⁻², with ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻². The 4π is not decoration; it belongs to the surface area of a sphere, and it is parked here so that it cancels later when you meet Gauss's law. The sheer size of k is telling you that the coulomb is an enormous unit of charge. Two 1 C point charges 1 m apart would push each other with 9.0 × 10⁹ N, the weight of some nine hundred thousand tonnes. Nothing on a bench comes close: raise both charges to 10 nC and set them 5.0 cm apart and the push is still only 3.6 × 10⁻⁴ N, far too small to see without a torsion fibre. The same constant sets the contrast with gravity. In a hydrogen atom, with the electron 5.29 × 10⁻¹¹ m from the proton, the electric attraction is 8.24 × 10⁻⁸ N while the gravitational attraction between them is 3.6 × 10⁻⁴⁷ N, smaller by 2.3 × 10³⁹. Gravity rules the sky only because bulk matter is so nearly neutral that its charges cancel.

There are two ways to run a Coulomb calculation, and mixing them is where marks are lost. The safe route for hand work is to feed absolute values into F = k|q₁||q₂|/r², get a positive number, and decide the direction separately by inspection: like signs repel, unlike signs attract, and the force lies along the joining line. The formal route keeps the signs. Write F₂₁ = (kq₁q₂/r²)r̂₂₁, where F₂₁ is the force on charge 2 from charge 1 and r̂₂₁ is the unit vector pointing from charge 1 towards charge 2. If q₁q₂ is positive the force on charge 2 runs along r̂₂₁, away from charge 1; if q₁q₂ is negative the whole vector reverses and points back at charge 1. Run it on a one-dimensional case to watch the machinery bite: put q₁ = −3.0 nC at x = 0 and q₂ = +2.0 nC at x = +4.0 cm. Here r̂₂₁ = (+1, 0) and q₁q₂ = −6.0 × 10⁻¹⁸ C², a negative product, so F₂₁ points along −x̂, back towards the origin, which is the attraction the opposite signs demand. Its size is 8.988 × 10⁹ × (3.0 × 10⁻⁹)(2.0 × 10⁻⁹) ÷ (0.040)² = 3.4 × 10⁻⁵ N, and the magnitude route returns that same 3.4 × 10⁻⁵ N with the direction supplied by inspection. The two routes must always agree; what they must never do is both contribute a minus sign.

Once you have one force at one separation, stop recomputing from scratch. The constant and both charges cancel in a ratio: F₂/F₁ = (r₁/r₂)². If a fixed pair of charges exerts 45 µN at 6.0 cm, then at 15 cm it exerts 45 × (6.0/15)² = 45 × 0.16 = 7.2 µN, and no value of k or q was needed. Changes to the charges multiply in independently: double one charge and triple the separation and the force becomes 2 × (1/3)² = 2/9 of what it was. The same structure appears as a straight line on a log-log plot. Taking logarithms of F = kq₁q₂r⁻² gives ln F = ln(kq₁q₂) − 2 ln r, so a graph of ln F against ln r has gradient exactly −2 and an intercept that measures the charge product. That is how force-against-separation data are really tested: the gradient checks the exponent, and it does so without knowing either charge.

The symmetry is worth looking at directly. Coulomb's law contains the product q₁q₂ and a single separation r, and neither of those changes when you swap the labels 1 and 2, so the law is incapable of returning unequal forces: Newton's third law comes out of it as a theorem rather than going in as an extra assumption. What the symmetry does not control is the motion, because acceleration is F/m and nothing obliges the two masses to match. Take the hydrogen atom again. That same 8.24 × 10⁻⁸ N acts on the proton and on the electron, yet the electron accelerates at 9.1 × 10²² m s⁻² while the proton manages only 4.9 × 10¹⁹ m s⁻². The ratio of those accelerations is 1836, exactly the proton-to-electron mass ratio, which is why the standard picture has the electron sweeping around a proton that barely stirs. Read a Coulomb answer as one number shared by two bodies, then hand it to F = ma twice.

The law is written for points, so ask every time whether the objects qualify. Two spheres whose charge is spread uniformly and held in place act exactly like points at their centres, at any separation, by the same shell argument that lets you treat the Earth's gravity as though all its mass sat at the centre. Conductors are the awkward case, because each sphere's charge is free to move: the near faces draw together and the far faces retreat, so each sphere polarises the other. For conducting spheres of radius a at separation r carrying charges of comparable size, the leading fractional correction is about 4(a/r)³, and it always acts the same way — extra attraction, or less repulsion, never more. Two 5.0 mm spheres 40 mm apart are off by about 0.8%, small enough to ignore in a teaching lab; bring them to 15 mm apart and that leading term alone is 15%, with higher terms on top of it. Push two like-charged conductors close enough and induction can even turn the repulsion into an attraction, which the point model can never reproduce. A working rule is r ≥ 10a, which holds the leading term under half a percent — and the model assumes the charges are at rest, since moving charges bring magnetic forces and a signal delay Coulomb's law knows nothing about.