Intuition The one core ideaThe whole plum-pudding story is really one sentence: spread the positive charge evenly through a ball, drop an electron inside, and it will always be pushed back toward the centre — so it wiggles in place like a mass on a spring. Everything below is just the vocabulary and pictures you need to see why that push-back happens and how fast the wiggle is.
This page assumes you know nothing . We build every letter, arrow, and squiggle the parent note 1.2.3 throws at you, in an order where each idea leans only on the ones before it.
Definition Electric chargeCharge is a property that some tiny bits of matter carry that makes them push or pull on each other without touching. There are two flavours: positive (+ negative (− repel (push apart), opposite flavours attract (pull together).
The picture: think of two magnets, but instead of "north/south" we say "plus/minus".
Intuition Why the topic needs thisThomson's puzzle was "atoms are neutral yet contain negatives." That sentence is meaningless until you know that positive and negative charges cancel : put + e − e nothing . That cancellation is the entire reason the model has a "pudding" (positive) and "raisins" (negative).
Definition The electron and the lettere electron is the tiny negative particle Thomson discovered in 1897 (see Discovery of the electron (Thomson's cathode ray tube) ). The letter e size of its charge — a fixed positive number, e = 1.6 × 1 0 − 19 − e + e
So e name for a number , the smallest chunk of charge nature hands out. When you see − e + e
e Why it feels right: electrons are negative. The fix: by convention e positive magnitude (e > 0 separately as − e + 6 e
Definition Coulomb (C) and Coulomb's constantThe coulomb is the unit we measure charge in, like "metre" measures length. The strength of the push between two charges is set by a fixed number of nature written 4 π ε 0 1 = 9.0 × 1 0 9 ε 0 permittivity of free space — just a constant baked into how strong electric forces are in empty space.
You never need to compute ε 0 whole clump 4 π ε 0 1 9.0 × 1 0 9 charge-and-distance into force .
Intuition Why the topic needs itEvery force and field formula in the derivation opens with 4 π ε 0 1
Definition Electric fieldE this charge?", we ask "how strong is the electric push per coulomb at this spot?" That per-coulomb push is the electric field , written E q E F = q E
The picture: imagine invisible arrows filling space, one at every point, saying "a + 1 this hard, that way." Long arrow = strong field.
Intuition Why not just use force everywhere?Because the field is a property of the source (the pudding), independent of who we drop in. We compute E F = ( − e ) E
R r R fixed radius of the whole atom — the size of the pudding ball, about 1 0 − 10 r variable distance of the electron from the centre, from 0 R R r
Read R r
R r Why it feels right: they look almost the same letter. The fix: R r F = − 4 π ε 0 R 3 e 2 r except r r
Intuition Why this mattersThe positive charge is spread evenly , so charge is proportional to volume: the charge inside radius r volume of whole ball volume inside r = R 3 r 3 r 3 / R 3 the geometric fact that flips the usual 1/ r 2 E ∝ r
Definition Gauss's law, in wordsGauss's law says: to find the field at distance r pretend all the charge inside radius r r The charge in the outer shell contributes nothing to the field at r
The picture: draw an imaginary sphere ("Gaussian surface") through the electron. Only the charge trapped inside your imaginary sphere tugs it.
this tool
We need the field inside a charge cloud, where naive Coulomb ("1/ r 2 q in E = 4 π ε 0 1 r 2 q in
Worked example Watch the two facts combineEnclosed charge q in = e R 3 r 3 E = 4 π ε 0 1 r 2 q in = 4 π ε 0 1 R 3 e ⋅ r 2 r 3 = 4 π ε 0 1 R 3 e r r 3 1/ r 2 r That is why E ∝ r , growing from zero at the centre — not the 1/ r 2
Definition A restoring forceA restoring force is one that always points back toward a home position . Written F = − k r r minus says "the force points opposite to the displacement" — outward push → inward force. The number k spring constant ) sets the stiffness.
The picture: a mass on a spring. Pull it right, spring pulls left; push it left, spring pushes right. Always home-ward.
The parent's Step 3 lands on F = − 4 π ε 0 R 3 e 2 r F = − k r same shape . So the electron in the pudding behaves exactly like a mass on a spring, with k = 4 π ε 0 R 3 e 2
ω ν 2 π F = − k r oscillates , sliding back and forth through the centre forever. This is Simple Harmonic Motion (SHM). We describe how fast with:
ω angular frequency , radians of phase per second: ω = k / m ν ordinary frequency , full back-and-forth cycles per second (hertz): ν = 2 π ω π ≈ 3.14159 2 π
Intuition Why bother with a frequency at allA wiggling charge radiates light at its wiggle frequency. Thomson computed ν ∼ 1 0 15 ν
ω ν Why it feels right: both mean "how fast it repeats." The fix: they differ by exactly 2 π ω ν 2 π
neutral means they cancel
symbol e is a fixed number
Coulomb constant 1 over 4 pi eps0
enclosed charge as r cubed over R cubed
Gauss law only inside counts
Hooke law restoring force
omega and nu the light frequency
Cover the right side and test yourself — you are ready when each is instant.
What do the signs + − They repel (push apart); opposite signs attract.
What does "neutral atom" mean in charge terms? Equal positive and negative charge, so they cancel and outside feels nothing.
Is the symbol e Positive — it is the magnitude ; the electron's charge is − e
What single number is 4 π ε 0 1 9.0 × 1 0 9
Given field E q F = q E
Difference between R r R r
Why is the charge inside radius r e r 3 / R 3 Charge is spread evenly, so it scales with volume, and volume ∝ r 3
What does Gauss's law let you ignore? All charge outside radius r
Why is the field inside the pudding ∝ r 1/ r 2 Enclosed charge ∝ r 3 1/ r 2 r
What does the minus sign in F = − k r The force points opposite the displacement — always back toward the centre.
How are ω ν ν = ω / ( 2 π )
Why do we even compute the oscillation frequency? A wiggling charge radiates light at that frequency; Thomson hoped it matched atomic spectra.
