van der Waals forces — London dispersion, dipole-dipole, dipole-induced dipole: Foundations

Before you touch the parent note, you need a toolbox. This page names every symbol and idea the parent quietly assumes, gives each a plain-words meaning, a picture, and a reason the topic needs it. Read top to bottom — each block is built from the one above.

1. Charge: the raw ingredient — $q$

The one rule to remember: opposite signs attract, same signs repel. Nothing in this whole topic breaks that rule — it just dresses it up.

Figure 1 below draws that rule directly: the arrows on the top pair (labelled "attract") point inward, pulling the opposite charges together; the arrows on the bottom pair (labelled "repel") point outward, pushing the like charges apart. Read the arrow directions, not the colours — inward means attract, outward means repel.

Figure — van der Waals forces — London dispersion, dipole-dipole, dipole-induced dipole

Figure 1 — Charge $q$ and its one rule. Top pair: opposite signs, arrows point inward = attract. Bottom pair: like signs, arrows point outward = repel.

2. Distance between charges — $r$

3. Scalars, vectors, and the unit vector — $\hat{r}$

Before Coulomb's law, we must separate two different questions a formula can answer: "how big?" and "which way?"

Definition Scalar vs vector, and the unit vector

$\hat{r}$

A scalar is just a size — a plain number with units (e.g. "the strength of the force is 5 newtons"). We write it in ordinary italics, like $F$ .
A vector is a size and a direction — an arrow (e.g. "5 newtons, pointing that way"). We write it in bold, like $F$ , and draw it as an arrow.
The ==unit vector $\hat{r}$ == (say "r-hat") is a special arrow of length 1 that carries only the direction from charge 1 to charge 2. It answers "which way?" and nothing else.
Why the topic needs it: Coulomb's law and the field come in two versions. The scalar version ( $F$ , $E$ ) tells you how strong; the vector version ( $F = F \hat{r}$ , $E$ ) glues that strength onto a direction with $\hat{r}$ . When the parent note talks about attraction vs repulsion, or dipole arrows lining up, it is really talking about the vector version. Keep the two apart: strength is a scalar, "attract or repel" is about the direction $\hat{r}$ points.

So whenever you see a bare $F = \dots / r^{2}$ below, read it as "the magnitude (size) of the force." Its direction — along $\hat{r}$ , and inward for opposite charges, outward for like charges — is the separate fact drawn in Figure 1.

4. Coulomb's law — the engine

Formula Coulomb's law (magnitude, then direction)

Two point charges $q_{1}$ and $q_{2}$ a distance $r$ apart have force magnitude and stored energy: $F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}}, U = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r}$ and the full vector force is that magnitude aimed along the line between them: $F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}} \hat{r}$

$F$ = force magnitude (how hard they push/pull, in newtons $N$ ); $F$ adds the direction via $\hat{r}$ .
$U$ = potential energy (energy stored in their arrangement, in joules $J$ ). Energy is a scalar — it has no direction — so it needs no $\hat{r}$ .
$ε_{0}$ = a fixed constant of nature ("permittivity of free space") that just sets the units; you never need its value here, only that it's a positive number.
Why the topic needs it: this is the parent equation. Every boxed formula in the topic (dipole–dipole, Debye, London) is Coulomb's law applied to charges that nearly cancel. See Coulomb's law.

5. The electron cloud — the "fuzzy" picture

Figure 2 shows exactly this: the red dot is the $+$ nucleus, and the purple haze is the $-$ electron cloud — darkest where the electron is most likely to be, fading outward.

Figure 2 — The electron cloud: a fuzzy, deformable haze of negative charge around the positive nucleus.

6. The dipole — separated charge — $d$ , $μ$

Definition Dipole and dipole moment

$μ$ A dipole is a pair of equal and opposite charges, $+ q$ and $- q$ , held a small distance $d$ apart. Its strength is the dipole moment: $μ = q d$

$d$ = distance between the two charges (a length, in $m$ ).
$μ$ = "how much charge, how far apart" — bigger $q$ or bigger $d$ ⇒ bigger $μ$ . Units: coulomb-metre, $C \cdot m$ (charge $\times$ distance — read the units straight off $μ = q d$ ).
Picture and direction convention: we draw $μ$ as an arrow whose tail sits on the $-$ charge and head points to the $+$ charge (the physics convention used throughout this topic; chemists sometimes draw it the other way, but we fix ours as $- \to +$ ). The arrow's length stands for the size of $μ = q d$ , and its direction stands for which way the separation points. So $μ$ is itself a vector (Section 3) — how strong (length) and which way (arrow), written $μ$ when we mean the arrow and $μ$ when we mean just its length.
Why the topic needs it: neutral molecules have zero net charge, so Coulomb's law between whole molecules gives zero. Attraction survives only because charge is separated into a dipole. $μ$ is the number (and direction) that measures that separation. Deep dive: Dipole moment and molecular polarity.

Three flavours of dipole the topic uses — name them now:

Name	When it appears	Lasts?
Permanent dipole	molecule is built lopsided (e.g. HCl)	always there
Induced dipole	a nearby field pushes the cloud off-centre	only while the field is present
Instantaneous dipole	electrons randomly bunch for a split second	flickers on and off

7. Polarisability — how floppy the cloud is — $α$

Definition Polarisability

$α$ ==Polarisability $α$ == measures how easily an electron cloud is distorted (pushed off-centre) by an electric field. Put a molecule in a field $E$ and the induced dipole it develops is: $μ_{ind} = α E$

Big $α$ = floppy, easily-squished cloud (big, many electrons, loosely held).
Small $α$ = stiff cloud (small, tightly-held electrons).
Units: rearrange $α = μ_{ind} / E$ to read the units off — dipole moment ( $C \cdot m$ ) divided by field ( $V/m$ , see next section) gives $C \cdot m^{2} / V$ .
Picture: the same field arrow squishing a big soft cloud a lot, a small tight cloud only a little.
Why the topic needs it: induced-dipole and London forces are entirely governed by $α$ . "More electrons ⇒ stronger dispersion" is really "more electrons ⇒ bigger $α$ ." See Polarisability and Fajans' rules.

Figure 3 puts a big floppy cloud (left) and a small stiff cloud (right) in the same field arrows: the left cloud's $+$ centre shifts far, the right one barely moves — that difference is $α$ . (The arrows are all identical in length: same field, different squish.)

Figure 3 — Polarisability $α$ : the same field $E$ squishes a floppy cloud a lot (large $α$ , $+$ centre shifts far) but a stiff cloud only a little (small $α$ ).

8. The electric field — the messenger — $E$

Definition Electric field

$E$ An ==electric field $E$ == is the push per unit charge that a charge or dipole projects into the space around it. It is a vector $E$ : at each point it has both a strength and a direction (drawn as an arrow).

Units: newtons per coulomb, $N/C$ , equivalently volts per metre, $V/m$ — literally "how many newtons of force each coulomb of charge would feel here."
Link back to Coulomb's law: the force on a charge $q$ sitting in a field is (magnitude) $F = q E$ , or in full $F = q E$ . Compare with Coulomb's law: peel off the visiting charge $q_{2}$ and what remains, $E = \frac{1}{4 π ε _{0}} \frac{q _{1}}{r ^{2}}$ , is the field that charge $q_{1}$ makes. So $E$ is just "Coulomb's law with the second charge removed" — the reach a charge offers before a partner arrives.
Picture: arrows radiating out from a dipole, longer (stronger) near it, shorter far away — and, as Section 6 warned, lumpier in some directions than others.
Why the topic needs it: one molecule's dipole talks to another through its field. The parent writes $E_{dipole} \propto μ / r^{3}$ ; that is the field magnitude carrying the dipole's influence outward and fading as $1/ r^{3}$ . Multiply that field by a visiting charge or dipole ( $F = q E$ , or $μ_{ind} = α E$ ) to turn "reach" into an actual force.

9. Temperature and thermal tumbling — $T$ , $k_{B}$

10. Ionisation energy — how tightly held — $I$

11. Proportionality — reading $\propto$ and $1/ r^{6}$

Figure 4 plots $1/ r$ , $1/ r^{3}$ and $1/ r^{6}$ on the same axes so you can see the difference: at $r = 2$ the steepest ( $1/ r^{6}$ ) curve has already collapsed to $1/64$ of its start, while the shallowest ( $1/ r$ ) curve is only halved. Each curve is also labelled directly on the plot, so you needn't rely on colour.

Figure 4 — Why the exponent matters: a bigger power of $1/ r$ fades far faster with distance. At $r = 2$ only $1/64$ of the $1/ r^{6}$ strength remains.

12. How it all feeds the topic

How to read this diagram: each box is one idea from this page (or, on the far right, the parent topic). An arrow "A → B" means "you need A before B makes sense." Follow the arrows left-to-right: the raw ingredients (charge, distance) flow into Coulomb's law, which builds the dipole and field, which — combined with polarisability, temperature and ionisation energy — feed the three van der Waals forces on the right.

If the diagram below appears as raw text (some renderers do not draw Mermaid), read it as this plain chain instead:

charge $q$ , distance $r$ , $ε_{0}$ , and the scalar/vector distinction → Coulomb's law ( $U$ , $F$ , $F$ ).
Coulomb's law + electron cloud → dipole $μ = q d$ → field $E \propto μ / r^{3}$ (direction-dependent).
electron cloud → polarisability $α$ .
field + temperature $T, k_{B}$ → dipole–dipole ( $U_{dd}$ , Keesom).
field + polarisability → dipole–induced dipole (Debye).
polarisability + ionisation energy $I$ → London dispersion.
all three → van der Waals forces, tail $\propto 1/ r^{6}$ .

Everything on the left is what this page taught you; everything on the right is the parent note. If any left-hand box felt shaky, reread its section before opening the parent topic.

Equipment checklist

Cover the right side and answer out loud before revealing.

What does

q

mean, its units, and its one rule?

Electric charge (in coulombs, C); opposite signs attract, same signs repel.

What is

r

and its units?

The distance between two charges or molecules, centre to centre, in metres (m).

What is the difference between a scalar and a vector?

A scalar is a size only (a number); a vector is a size and a direction (an arrow).

What does the unit vector

\hat{r}

carry?

Only direction — an arrow of length 1 pointing from charge 1 to charge 2.

State Coulomb's law for energy.

U

equals

q_{1} q_{2}

divided by

4 π ε_{0} r

; energy falls off as

1/ r

and is a scalar (no direction).

Does a negative

U

mean attraction or repulsion?

Attraction — a bound, low-energy state.

Does energy

U

fall off faster or slower with

r

than force

F

Slower —

U \propto 1/ r

while

F \propto 1/ r^{2}

What is an electron cloud?

A smeared-out haze of negative charge around the nucleus, more likely where denser.

Define a dipole and its moment

μ

, with units and arrow direction.

Two opposite charges

+ q, - q

a distance

d

apart;

μ = q d

C \cdot m

, drawn as an arrow from

-

+

Is a dipole's field the same in every direction?

No — it is direction-dependent (lumpy); the

1/ r^{3}

only gives the distance fade, not the angular shape.

Name the three flavours of dipole.

Permanent, induced, and instantaneous.

What does polarisability

α

measure, its formula and units?

How easily a cloud is distorted;

μ_{ind} = α E

, units

C \cdot m^{2} / V

What is the electric field

E

, its units, and its link to force?

The push per unit charge (vector, in N/C or V/m); the force on a charge is

F = q E

What does the subscript in

U_{dd}

mean?

Dipole–dipole — the potential energy stored between two permanent dipoles.

What role does

k_{B} T

play, and its units?

The typical thermal jiggle energy (in joules;

k_{B}

in J/K,

T

in K) that scrambles dipole alignment.

What does

I

(ionisation energy) tell us?

How tightly electrons are held; low

I

means bigger instantaneous dipoles.

By how much does

U \propto 1/ r^{6}

shrink if

r

doubles?

2^{6} = 64

times.