Fields of Point Charges

Coulomb's law gives the force on a test charge q₀ sitting a distance r from q: F⃗ = k q q₀ r̂/r². Divide by q₀ and the probe vanishes from the result, leaving E⃗ = k q r̂/r². That division is the whole move, and it is why the field is a property of the source and of the point rather than of whatever happens to be sitting there. Formally the limit q₀ → 0 is taken, so the probe cannot push the source around while it measures. Put numbers through it: for q = 1.0 nC at r = 3.0 cm, E = (8.99 × 10⁹ × 1.0 × 10⁻⁹) ÷ (0.030)² = 8.99 N m² C⁻¹ ÷ 9.0 × 10⁻⁴ m² = 9.99 × 10³ N C⁻¹. Nothing in that calculation knows whether anything is sitting at 3.0 cm.

The symbol r̂ is not one fixed direction. It is a different unit vector at every field point, always running from the charge outwards to the point you asked about, so the arrows of a single point charge fan out along radii and do nothing else. Keep the sign inside q and the direction looks after itself: q > 0 gives E⃗ along +r̂, pointing away; q < 0 gives −r̂, pointing back towards the charge. The magnitude is k|q|/r² either way. Take q = +1.0 nC at the origin and the point (3.0 cm, 4.0 cm): r = 5.0 cm, so r̂ = (0.60, 0.80) and E = 3.60 × 10³ N C⁻¹, giving E⃗ = (2.16 × 10³, 2.88 × 10³) N C⁻¹. Change the charge to −1.0 nC and both components reverse while the magnitude is untouched.

When the charge sits at r⃗ₛ rather than at the origin, the separation vector is r⃗ − r⃗ₛ and the unit vector is r̂ = (r⃗ − r⃗ₛ)/|r⃗ − r⃗ₛ|. Substituting that into E⃗ = kq r̂/|r⃗ − r⃗ₛ|² produces the cube: E⃗ = kq (r⃗ − r⃗ₛ)/|r⃗ − r⃗ₛ|³. Count the powers before believing the exponent. The denominator carries three powers of length, but the numerator is a vector whose own length is that same distance, so two powers survive and the magnitude still falls as 1/r². Check it arithmetically with a charge at the origin and a field point at (0.060 m, 0.080 m): the numerator has length 0.100 m, the denominator is 0.100³ = 1.00 × 10⁻³ m³, and the ratio is 1.00 × 10² m⁻², which is exactly 1/(0.100 m)².

You rarely need E twice; you need to know how E changed. The ratio form does that without k ever appearing: E₂/E₁ = (r₁/r₂)². For a 1.0 nC charge, moving from 2.0 cm to 4.0 cm takes the field from 2.25 × 10⁴ N C⁻¹ to 5.62 × 10³ N C⁻¹, a factor of exactly four for a factor of two in distance. Take logarithms and the same law becomes log E = log(k|q|) − 2 log r: on log–log axes it is a straight line of slope −2, and changing q slides that line up or down without tilting it, which is a clean way to see that the charge sets the strength while the geometry sets the fall-off. Differentiate instead and you get dE/dr = −2E/r, so dE/E = −2 dr/r: a 1% error in your distance is a 2% error in the field.

As r → 0 the formula sends E to infinity, and that is a statement about the model, not about nature. Real charge lives on objects with size. For a sphere carrying its charge uniformly, the field outside is exactly kQ/r² measured from the centre, so the point model holds right down to the surface and only fails inside — which is where the formula's floor really sits. The surrounding air adds a second floor: dry air breaks down near 3.0 × 10⁶ V m⁻¹, so a 1.0 nC charge cannot be squeezed inside a radius of √(k|q|/E) = √(8.99 × 10⁹ × 1.0 × 10⁻⁹ ÷ 3.0 × 10⁶) = 1.7 mm without sparking. For fundamental particles, classical electrostatics is replaced long before the divergence bites: a proton's field at the Bohr radius, 5.29 × 10⁻¹¹ m, is already 5.15 × 10¹¹ N C⁻¹.

Go the other way and the approximation improves. Model a small lump of total charge q as two charges q/2 a distance d apart, and measure on the perpendicular bisector: each contributes k(q/2)/(r² + d²/4), the components across the bisector cancel, and the sum is E = kq r/(r² + d²/4)³⁄² = (kq/r²)(1 + d²/4r²)^(−3/2). At r = 10d the true field is 0.37% below the point-charge value, and at r = 6d it is 1.0% below. Measured along the line through the two charges the correction has the opposite sign, 0.75% high at r = 10d, but it shrinks just as fast, as (d/r)². So far means a few times the source's own size, and beyond that a point charge is an excellent model of anything carrying net charge.