Electric Field — E = F/q, Field Lines and Worked Examples

Q: Which way do field lines point?

By convention, the direction a *positive* test charge would be pushed. Field lines therefore stream out of positive charges and into negative ones.

High School⏱ 10 min readElectricity & Magnetism

The electric field: force per unit charge. Field lines, uniform fields, E = F/q and E = kQ/r², with worked examples.

Quick answer

The electric field at a point is the electric force that a tiny positive test charge would feel there, divided by that charge: E = F/q. It lets us describe electrical influence filling space, without action at a distance.

$\vec{E} = \frac{\vec{F}}{q}, \qquad E = k\frac{Q}{r^2}$

Unit: N/C or V/m

Example: A 2 µC charge feels 0.06 N in some field, so E = 0.06/(2×10⁻⁶) = 30,000 N/C at that point — pointing away from the positive source.

What you will learn

Define the electric field as force per unit charge
Use E = F/q and E = kQ/r² in problems
Interpret field-line diagrams correctly
Describe the uniform field between parallel plates

Action at a distance, explained

How does one charge push another across empty space? Physicists answer with the field: every charge fills its surroundings with an electric field, and other charges respond to the field where they are.

The field turns 'spooky action at a distance' into local cause and effect: charge A creates field E everywhere; charge B feels force F = qE from the field at B's location. Two steps instead of one mysterious leap.

Defining the field: E = F/q

The electric field $\vec{E}$ at a point is defined as the force per unit positive test charge:

$\vec{E} = \frac{\vec{F}}{q}$

Because we divide by the test charge, $\vec{E}$ doesn't depend on it — it describes the influence present at that point, waiting for any charge to feel it. Direction: the way a positive test charge would be pushed.

\vec{E} = \frac{\vec{F}}{q}

Vector: E points along the force on a positive test charge

Field of a point charge

Combine $E = F/q$ with Coulomb's law and the test charge cancels, leaving the field of a point charge $Q$ :

$E = k\frac{Q}{r^2}$

Same inverse-square falloff, pointing radially outward from positive $Q$ (inward for negative). Superposition applies: the total field from many charges is the vector sum of the individual fields — computed component by component.

Reading field-line diagrams

Field lines visualize E with four strict rules:

Lines point in the direction of $\vec{E}$ (out of +, into −).
Line density shows field strength — crowded lines mean strong field.
Lines never cross (the field has one direction at each point).
Lines begin on positive charges (or infinity) and end on negative charges (or infinity).

Between two oppositely charged parallel plates, the lines are straight, parallel, equally spaced — a uniform field, the workhorse of capacitors.

Worked mini-example

A point charge $Q = +5.0\ \mathrm{nC}$ sits alone. What is the field magnitude $0.10\ \mathrm{m}$ away?

$E = k\frac{Q}{r^2} = (8.988 \times 10^9)\frac{5.0 \times 10^{-9}}{0.10^2}$

$E = (8.988 \times 10^9)(5.0 \times 10^{-7}) \approx 4494\ \mathrm{N/C}$

About $4.5 \times 10^3\ \mathrm{N/C}$, directed radially outward. A 1 C test charge there would feel 4494 N — but real test charges are microcoulombs, so real forces are millinewtons.

Key idea

Field (E) is the landscape; force (F = qE) is what a particular charge feels walking through it. Keep them distinct and electrostatics becomes bookkeeping.

Variables and units

Symbol	Name	Unit	Meaning
$E$	Electric field	N/C	Force per unit charge at a point in space
$q$	Test charge	C	Charge experiencing the force
$Q$	Source charge	C	Charge creating the field
$r$	Distance	m	Distance from the source charge

Common mistakes

✗ Field lines are physical objects in space.

They're a visualization tool — like contour lines on a map. The field itself is continuous; lines just show its direction and relative strength (denser = stronger).

✗ The electric field is the same thing as electric force.

Force depends on the charge you place there (F = qE); the field E exists regardless. The field is the stage, the force is what a charge experiences on it.

✗ The field only exists where a test charge sits.

The field fills space around every charge, test charge or not. E = F/q defines it via a hypothetical probe; the field doesn't need the probe to exist.

Real-world applications

Capacitors store energy in the uniform field between their plates — the basis of camera flashes and defibrillators.
Old CRT televisions steered electron beams with electric fields to paint the picture, pixel by pixel.
Particle accelerators use precisely shaped electric fields to push charged particles to near light speed.

Frequently asked questions

What is the unit of electric field?

Newtons per coulomb (N/C) — equivalently volts per meter (V/m). Both describe the same thing: how much force (or voltage drop) per unit charge (or meter).

Which way do field lines point?

By convention, the direction a positive test charge would be pushed. Field lines therefore stream out of positive charges and into negative ones.

What is a test charge?

An imaginary tiny positive charge used to probe the field without disturbing it. The field is defined as the force per unit charge, so the test charge's size cancels out.

What is the electric field inside a conductor?

Zero. Free charges in a conductor rearrange until they cancel any internal field — which is why a car (a metal shell) protects you from lightning: the field stays outside.