The Electric Field

Rub a balloon on your hair and hold it near a thin stream of water from a tap. The stream bends toward the balloon — visibly, silently, across a gap of empty air. Nothing touches the water; no string pulls it. So what reaches across that gap and tugs? For two centuries physicists were quietly uneasy about this. Newton's gravity had the same problem and he hated it: how can one body reach out and shove another with nothing in between? "Action at a distance," he wrote, is "so great an absurdity that I believe no man… can ever fall into it."

The cure — one of the most powerful ideas in all of physics — is the field. We stop thinking of the balloon reaching all the way over to the water. Instead we say the balloon's charge fills the space around it with a condition, an invisible influence present at every point, called the electric field. The water doesn't feel the distant balloon at all; it feels only the field right where it sits. Every interaction becomes local: a charge makes a field everywhere, and any other charge simply responds to the field at its own location. This page is about that one idea — what the electric field is, how to compute it, and how to picture it.

Definition: force per unit charge

To measure the field somewhere, do the obvious experiment: place a tiny positive test charge q_0 at that point and see what force it feels. Double the test charge and the force doubles; triple it and the force triples. The force depends on how much charge you brought — but the force per unit charge does not. That ratio is a property of the point in space itself, set up by whatever source charges are lurking nearby. We call it the electric field \vec{E}.

Definition. The electric field at a point is the force per unit positive test charge placed there: \vec{E} = \frac{\vec{F}}{q_0}.
It is a vector field. At every point of space it has a magnitude and a direction — an arrow attached to each location. Run the definition backwards and a charge q dropped into a field \vec{E} feels \vec{F} = q\vec{E} (a negative charge feels a force opposite to the field).
Units. Newtons per coulomb, \text{N/C} — which turns out to be identical to volts per metre, \text{V/m}, the form you'll meet again once potential arrives.

The field of a single point charge

Now put a source charge Q at the origin and ask what field it makes at a distance r. We already know the force on a test charge q_0 from Coulomb's law: F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q\,q_0}{r^2}. Divide by q_0 and the test charge vanishes from the answer, exactly as it should — the field belongs to Q alone.

Magnitude. With k = \dfrac{1}{4\pi\varepsilon_0} \approx 8.99\times 10^{9}\ \text{N·m}^2/\text{C}^2, \vec{E} = \frac{kQ}{r^2}\,\hat{r} = \frac{Q}{4\pi\varepsilon_0 r^2}\,\hat{r}.
Direction. The unit vector \hat{r} points from the charge outward. So the field points away from a positive Q and toward a negative one.
Inverse square. Move twice as far away and the field drops to a \tfrac{1}{4}; three times as far, a \tfrac{1}{9}.

A picture makes the "arrow at every point" idea concrete. Below, a positive charge sprays its field radially outward; drop a small test charge onto one of those arrows and it is pushed straight along it. Reveal the figure step by step.

Field lines: a map of the field

Drawing an arrow at every point quickly becomes a thicket. Michael Faraday — who could barely do the algebra but saw physics in pictures better than anyone before or since — invented a cleaner way: field lines (he called them "lines of force"). Instead of countless little arrows, draw continuous curves that thread along the field. They obey a short, strict set of rules:

Direction. The field at any point is tangent to the line through it — the line's arrowhead tells you which way \vec{E} points.
Start and finish. Lines begin on positive charges and end on negative charges (or run off to infinity). Positive charges are sources; negative charges are sinks.
Density is strength. Where lines crowd close together the field is strong; where they spread apart it is weak. Near a point charge they fan out with distance, which is the 1/r^2 weakening drawn as a picture.
They never cross. If two lines met, the field would point two ways at once at that spot — impossible. A field has exactly one direction at each point.

Superposition: fields just add

What if several charges are present at once? Here nature is kind. Each source makes its own field as if the others weren't there, and the total field is the vector sum of them all:

\vec{E}_{\text{tot}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \cdots = \sum_i \frac{kQ_i}{r_i^2}\,\hat{r}_i.

Add the arrows tip-to-tail, not the numbers — direction matters. For a smooth, continuous smear of charge (a charged rod, a disc, a sphere) the sum becomes an integral: chop the object into infinitesimal pieces dq, each contributing a tiny field, and add them up:

\vec{E} = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r^2}\,\hat{r}.

You will grind through such integrals soon enough; the point for now is that nothing new is going on — it is superposition all the way down. Two of these results are worth carrying in your pocket, because they show the field of a shape can fall off completely differently from a point charge's.

Infinite line charge with linear charge density \lambda (C/m), at perpendicular distance r: E = \frac{\lambda}{2\pi\varepsilon_0 r} \qquad (\text{falls off as } 1/r).
Infinite plane sheet with surface charge density \sigma (C/m²): E = \frac{\sigma}{2\varepsilon_0} \qquad (\text{constant — independent of distance!}).

Stack these side by side and a pattern leaps out. Pile charge into a point and its field dies as 1/r^2. Smear it along a line and the field dies more gently, as 1/r. Spread it over an infinite plane and the field doesn't die at all — it is the same a millimetre away as a kilometre away. More symmetry, slower fall-off. The chart below plots all three magnitudes together.

Worked examples

Example 1 — field of a point charge. A small sphere carries Q = 5.0\ \text{nC} = 5.0\times 10^{-9}\ \text{C}. What is the field magnitude 0.20\ \text{m} away?

E = \frac{kQ}{r^2} = \frac{(8.99\times 10^{9})(5.0\times 10^{-9})}{(0.20)^2} = \frac{44.95}{0.040} \approx 1.1\times 10^{3}\ \text{N/C},

pointing radially away from the sphere (it's positive). A test charge dropped there feels F = q_0 E along that outward direction.

Example 2 — two charges, net field at the midpoint. Place +Q and -Q a distance 2a apart (a dipole) and stand at the midpoint. The field from +Q points away from it — toward the negative side. The field from -Q points toward it — the same way. They reinforce, and each has magnitude kQ/a^2, so

E_{\text{mid}} = \frac{kQ}{a^2} + \frac{kQ}{a^2} = \frac{2kQ}{a^2},

pointing from the + charge toward the - charge. Contrast: if instead both charges are +Q, the two fields at the midpoint point in opposite directions with equal size and cancel exactly — E_{\text{mid}} = 0. Same geometry, opposite outcome, all decided by vector superposition.

Example 3 — the line-charge formula. A very long wire carries \lambda = 4.0\ \text{nC/m}. What is the field 0.10\ \text{m} from it? Using \dfrac{1}{2\pi\varepsilon_0} = 2k,

E = \frac{\lambda}{2\pi\varepsilon_0 r} = \frac{2k\lambda}{r} = \frac{2(8.99\times 10^{9})(4.0\times 10^{-9})}{0.10} \approx 7.2\times 10^{2}\ \text{N/C}.

Step back to twice the distance, 0.20\ \text{m}, and this halves (not quarters) — the tell-tale 1/r signature of a line.

The field is not made by the test charge, and it doesn't need one to exist. The definition \vec{E} = \vec{F}/q_0 uses a test charge only as a probe — a way to reveal a field that is already there. Take the probe away and the field of the source charges is completely unchanged; empty space around a charge is genuinely filled with field, whether or not anything is there to feel it. Thinking "no test charge, no field" is like thinking a hill has no slope until you roll a ball down it.

There is a subtlety hidden in that little word "test", too. We divide by q_0 and imagine it as small — and it genuinely must be. A real probe charge pushes back on the source charges (or shifts charge around in a nearby conductor), disturbing the very field you're trying to measure. So the honest definition is a limit: \vec{E} = \lim_{q_0 \to 0}\vec{F}/q_0 — a test charge small enough to look without touching.

One more classic slip: field lines are not the paths particles travel. A released charge accelerates along the field, but it also builds up speed, so its actual trajectory curves away from the line as it goes. A field line shows the direction of the force at each point, not the route of any journey.

Michael Faraday left school at thirteen and taught himself science while binding books. He never mastered calculus, and the equation-wielding physicists of his day were faintly patronising about his "lines of force" — pretty pictures, they thought, not real physics. But Faraday believed the lines were real: he pictured space itself as a taut, invisible fabric of field, and charges and magnets as things that stretch and reshape it. That intuition let him see connections the equation-people missed, culminating in his discovery of electromagnetic induction.

The punchline is delicious. Decades later James Clerk Maxwell took Faraday's homespun pictures, wrote them in mathematics, and produced the four equations that unified electricity, magnetism and light — the crown jewel of classical physics. Maxwell was generous about the debt: the field, he said, was Faraday's idea first. The apprentice who couldn't do the algebra had seen the deepest thing, and the field concept he championed is now the language of all modern physics, from radio to quantum field theory.