Electric field & the test charge

Coulomb's law is a statement about a pair: put one charge here and another there, and a force appears that is proportional to both of them. That is awkward bookkeeping once a dozen charges sit on a plate, because every new charge restarts the whole calculation. The field model breaks the statement in half. First, freeze the charges already present and call them the sources. Second, walk through the space around them with a small positive probe and record, at every point, the force per unit charge. That record is a vector at each point — a field — and it stands whether or not a probe happens to be there. The payoff is locality: to find the force on a charge placed at P you no longer need to know where the sources sit or how far away they are. You need one vector, the field at P, and the charge itself.

Dividing by q₀ is allowed because measurement says the force is strictly proportional to the probe's charge, so long as the sources stay put. Place +1.0 nC at P and read 3.0 × 10⁻⁵ N; replace it with +3.0 nC at the same point and the reading becomes 9.0 × 10⁻⁵ N; use −2.0 nC and the magnitude is 6.0 × 10⁻⁵ N, reversed. In every case F/q₀ = 3.0 × 10⁴ N C⁻¹. Coulomb's law already contains that proportionality, since the probe's charge enters to the first power, and superposition carries it to any number of sources: each source contributes a force proportional to q₀, and a sum of terms each proportional to q₀ is itself proportional to q₀. The ratio therefore drops the probe out of the answer, which is precisely what a definition of a property of the point requires. Had F gone as q₀², the ratio would still depend on the probe and no field could be defined this way.

The definition names a positive test charge, and that choice — convention, not physics — fixes the direction. E points the way a positive probe would be pushed. Put a negative probe at the same point and it is dragged the opposite way, yet nothing about the field has changed; only the sign of q in F = qE has. The two vectors are therefore parallel or antiparallel and never at any other angle, because q is a scalar and multiplying E by it can only stretch the vector and possibly reverse it. Concretely: if E at P is 2.0 × 10⁴ N C⁻¹ pointing east, a +5.0 nC charge at P feels 1.0 × 10⁻⁴ N east and a −5.0 nC charge feels 1.0 × 10⁻⁴ N west. Both readings report the same eastward field.

Two charges in a problem can have two quite different jobs. Sources make the field; the probe samples it. The distinction bites hardest in F = qE, where E must be the field produced by every charge except q itself: a charge exerts no net force on itself, and its own field diverges at its own location, so including it would make the expression meaningless. The field you draw around a source is the field that acts on other charges, never on the source that made it. The roles are not permanent either. The probe is itself a source, and by Newton's third law it pulls or pushes the source charges with a force of equal magnitude. Which charge you call source and which you call probe is a modelling decision about what you want to know — and the next section is about what that decision costs.

Every measurement of this kind is paid for. The probe attracts or repels the very charges that produce the field, with an equal and opposite force, and charges free to move will move; charge on a conductor always can. Take a source sphere on a mount that yields 1.0 mm per 100 μN of sideways force, 60.0 mm from a field point where the true field is 1.00 × 10⁵ N C⁻¹. A 2.0 nC probe pushes on it with 2.0 × 10⁻⁹ C × 1.00 × 10⁵ N C⁻¹ = 0.20 mN, which shifts it 2.0 mm. The probe now stands 62.0 mm away, where the field is weaker by the factor (60.0/62.0)² = 0.937, so the reading comes in about 6% low — an error the measurement created. Cut the probe to 0.20 nC and the shift is 0.20 mm and the error 0.7%. The disturbance shrinks with q₀, which is why the definition is written as a limit and not as a plain division.

The unit follows from the definition: newtons per coulomb. One newton per coulomb is also one volt per metre, since a volt is a joule per coulomb and a joule is a newton metre; the second name earns its keep when potential arrives, and the two are interchangeable everywhere. Three values are worth carrying. The fair-weather atmospheric field near the ground is about 1.0 × 10² N C⁻¹ pointing downward. Dry air breaks down into a spark near 3 × 10⁶ N C⁻¹, which caps the field any bench apparatus can hold. And at one Bohr radius from a proton, 5.29 × 10⁻¹¹ m, the field is 5.1 × 10¹¹ N C⁻¹. Nearly ten powers of ten separate the field you are standing in from the one that binds an atom, and knowing where a number falls in that range is the quickest check on an answer.