Intuition The one core ideaA charge quietly changes the space around it, storing at every point a ready-made "push per unit of charge" — that stored push is the electric field . Everything else on the parent page is just the toolkit for writing down that push as an arrow (magnitude + direction) and adding several such arrows together.
Before you can read the parent note, you need to already own a small pile of symbols and pictures. This page builds each one from absolute zero, in the order they lean on each other. If a symbol appears on the parent page, it is unpacked here first.
Definition Electric chargeq positive ( + ) negative ( − ) coulombs (symbol C Picture: a little dot with a + −
Why the topic needs it: charge is the source of everything. No charge, no field.
A rule that hands you a value at every point in space . A scalar field hands you one number (like temperature in a room). A vector field hands you a whole arrow (a size and a direction).
Picture: stand anywhere in a room; the room silently tells you a number (temperature) or an arrow (wind).
The figure below draws exactly this contrast. On the left half of the board, every grid point carries a single pale-yellow number — that is a scalar field (imagine reading the temperature at each spot). On the right half, every grid point instead carries a small blue arrow that has both a length and a heading — that is a vector field (imagine feeling the wind's strength and direction). The dotted vertical chalk line separates the two worlds. The electric field lives on the right: an arrow at every point.
Why the topic needs it: the electric field is a vector field. To even say "the field at that point" you must first accept that space can carry an arrow at every location.
The whole topic is arrows. So we need the vocabulary of an arrow.
Definition Vector and the arrow-hat vector is an arrow: it has a length (how strong) and a direction (which way). We put a little arrow over a letter to say "this is a vector": E F r Picture: an arrow drawn on paper. Its length is the strength; where its tip points is the direction.
Definition Magnitude — the same letter with no hatThe length of the arrow alone (a plain positive number, no direction). We write it with no arrow: E E F F Picture: measure the arrow with a ruler; that number is the magnitude.
^ ^ ^ along the positive x (to the right), ^ along the positive y (up). Any arrow is then "so much right plus so much up": A = A x ^ + A y ^ Picture: two little rulers glued to the floor and the wall; every arrow is measured against them.
r ^ point . Its length is exactly 1 direction and nothing else . The hat ^ r ^ r ^ ( x , y ) r ^ = r x ^ + r y ^ , r = x 2 + y 2 , ( x / r ) 2 + ( y / r ) 2 = 1 Picture: a stubby arrow of fixed length 1 E r ^ ×
Intuition Why split every arrow into (magnitude)× how strong (given by a formula in q which way (given by geometry). Writing E = E r ^
Why the topic needs it: the parent's headline formula E = 4 π ε 0 1 r 2 q r ^ × r ^ r ^ ^ , ^ x y
Definition Source point and field pointThe source point is where the charge sits (it makes the field). The field point is where you are asking "what's the field here?".
Picture: one dot is the charge (source); another dot is where your measuring probe sits (field point).
r Picture: a taut string stretched from the charge to your probe; its length is r
The figure below builds this triangle on the board. The pink dot at the origin is the source charge + q a y r r ^ from the charge toward the field point .
r ^ r ^ from the source charge toward the field point — the direction the string points as it leaves the charge. In this figure r ^ = r a ^ + r y ^ Picture: stand on the charge, look at the probe; r ^
Common mistake The single most commonr ^ r ^ toward the charge because attraction "feels" like pulling in.
Fix: r ^ sign of q does the attraction/repulsion, not r ^
Intuition From 2-D distance to a 3-D sphereRight now we drew the triangle flat on the board (a 2-D slice) just to compute the number r r all the points that far away form a whole sphere of radius r r r = x 2 + y 2 + z 2 4 π r 2
Why the topic needs it: every field formula ends in r ^
π ≈ 3.14159 sphere , and spheres are round.
Picture: a circle; π = 4 π r 2
ε 0 ε 0 ≈ 8.85 × 1 0 − 12 how strongly empty space responds to charge — think of it as the "stiffness of the vacuum". Its value converts charges-and-distances into real newtons.
Picture: a dial on the universe fixing the overall loudness of all electric effects.
Why the topic needs it: 4 π ε 0 1 every field and force formula on the parent page. It is a "spreads over a sphere" bookkeeping factor, nothing scarier.
Before we can define the field as "force per charge", we need the actual force. That force is Coulomb's Law .
Intuition Read the pieces you already ownEvery symbol here is one we just built: k q 1 q 2 1/ r 2 r ^
r 2 1/ r 2 r 2 r × r 1/ r 2 r − 2 r 1/ r 2 fast : double the distance, the field drops to a quarter .
Picture: the same handful of arrows (field lines) forced to cover a sphere. A sphere's area is 4 π r 2
The figure below makes the "spreading" literal. The pink charge at the centre fires out a fixed number of white field lines in all directions. Two chalk circles — a small blue one at radius r 2 r shells those lines must cross. The same lines cross both, but the outer shell has four times the area (area grows like r 2 is the inverse-square law.
1/ r 2 1/ r surface — and, as we insisted in §2, in real 3-D space that surface is a full sphere whose area grows like r 2 r 1/ ( area ) = 1/ r 2
Why the topic needs it: the entire strength of a point-charge field, and why field-line density means strength, both live in this 1/ r 2
F newtons ( N ) Picture: an arrow shoving the probe charge.
q 0 tiny positive charge we drop in just to feel the field. The subscript 0 Picture: a whisper-small +
lim q 0 → 0 q 0 → 0 lim Picture: shrinking the thermometer smaller and smaller so it stops warming the room it measures.
Why the topic needs it: this is the definition of the whole subject . Every other formula is this idea worked out for specific charges.
θ θ from the positive x ^ . So θ = 0 θ = 9 0 ∘ θ = 18 0 ∘ θ = 27 0 ∘ cos θ sin θ Picture: a protractor pinned at the origin with 0 ∘ x
E x E y shadow on the horizontal (x (E x shadow on the vertical (y (E y E = E x ^ + E y ^ Picture: shine a light straight down — the shadow on the floor is E x E y
cos θ sin θ E θ + x E x = E cos θ E y = E sin θ cos θ / sin θ / θ signs of E x , E y Picture: the arrow is the slanted side of a right triangle; cos θ
Why the topic needs it: in the parent's worked examples (b) and (c), the fraction cos θ = a 2 + y 2 y exactly one such projection. It's how the vertical parts survive and the horizontal parts cancel.
Definition The sum symbol∑ i ∑ i ( stuff i ) i stuff 1 + stuff 2 + … i Picture: a bucket you toss each charge's arrow into, then add them tip-to-tail.
Definition The integral symbol∫ d q ∫ ∑ Continuous Charge Distributions .)
Picture: the discrete bucket of arrows, but the dots become a smooth smear and the sum becomes a smooth total.
allowed to just add the arrows?
Because Coulomb's Law is linear in charge (double a charge, double its field) and arrows add tip-to-tail (Newton). That permission is the Superposition Principle — the third pillar of the parent page.
Here is how the pieces stack, read top to bottom : the raw materials (charge, arrows, distance) sit at the top; combining them builds Coulomb's force; dividing that force by the probe charge defines the field; and breaking fields into components lets us add many of them — which is the whole parent topic. (Each box is one plain-English idea; the arrows mean "is needed to build".)
Unit vector r-hat direction
Inverse square one over r squared
Point charge field formula
k equals one over four pi eps0
Force per test charge defines E
Electric field lines and superposition
Cover the right side and see if you can state each before revealing.
What does the hat in r ^ r ^ It marks a unit vector — a pure direction of length exactly 1
Which way does r ^ From the source charge to the field point (out of the charge).
Write r ^ x , y ( x , y ) r ^ = r x ^ + r y ^ with
r = x 2 + y 2 .
E E E is the whole arrow (size + direction);
E is just its length (magnitude).
State Coulomb's Law in full. F = k r 2 q 1 q 2 r ^ , with
r ^ from
q 1 to
q 2 ; sign of
q 1 q 2 sets attract/repel.
What is k k = 4 π ε 0 1 ≈ 8.99 × 1 0 9 N m 2 / C 2
Why does field strength go as 1/ r 2 The same field spreads over a 3-D sphere of area 4 π r 2 1/ r 2
What does E ≡ F / q 0 The force per unit test charge — strip out the probe's size to get a property of space.
Why the limit q 0 → 0 So the tiny probe never nudges the source charges and corrupts the field it measures.
From which axis is θ From the positive x
If an arrow of length E θ E x = E cos θ E y = E sin θ
What does ∑ i E i Add every charge's field arrow tip-to-tail (the superposition sum).
What does ∫ r 2 d q r ^ ∑ Adds infinitely many infinitesimal charge pieces of a continuous smear.
Why are we allowed to add fields at all? Coulomb's law is linear and forces add as vectors — the superposition principle.
See also: Coulomb's Law , Superposition Principle , Continuous Charge Distributions , Electric Dipole , and the parent topic note .
