Why Your Hair Stands Up After You Rub a Balloon on It

Rub a balloon on dry hair and the hair rises towards it; walk across a synthetic carpet and a door handle gives a tiny shock. Both are electric charges at work, and one simple law — Coulomb's law — explains the forces between them.

This part covers the properties of charge, Coulomb's law, the superposition principle, and electric fields and field lines.

**Electric charge is quantised, occurring only in whole-number multiples of $e=1.6\times 10^{-19}$ C, it is conserved in an isolated system, and it adds algebraically; conductors let charges move freely through them, while insulators hold charges where they are placed.

Properties:

- Quantisation** — $q=ne$, where $n$ is an integer
- Conservation — the total charge of an isolated system stays constant; charge is transferred, not created or destroyed
- Additivity — total charge is the algebraic sum, so $+3$ C, $-5$ C and $+1$ C give $-1$ C

Worked example — counting electrons. The number of electrons in a charge of $-1$ C is

$$n=\frac{q}{e}=\frac{1}{1.6\times 10^{-19}}=6.25\times 10^{18}$$

Conductors and insulators:

- Conductors — metals, the human body and the earth; excess charge spreads over the surface
- Insulators — glass, plastic and dry wood; charge stays where it is placed
- Earthing — connecting a charged conductor to the earth lets its excess charge flow away

An everyday example. A plastic comb run through dry hair picks up bits of paper because the charge stays on the insulating comb instead of draining away.

The substance. Quantisation is invisible in daily life — ordinary charges contain so many electrons that charge seems continuous.

**The force between two point charges is proportional to the product of the charges and inversely proportional to the square of their separation, $F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$, acting along the line joining them — repulsive for like charges and attractive for unlike ones.

Vector form.** The force on $q_2$ due to $q_1$ is

$${\vec{F}}_{21}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{21}^2}{\hat{r}}_{21}$$

where ${\hat{r}}_{21}$ points from $q_1$ to $q_2$, and ${\vec{F}}_{12}=-{\vec{F}}_{21}$.

Constants. $\frac{1}{4\pi {\epsilon }_0}=9\times 10^9$ N m${}^2$ C${}^{-2}$, with ${\epsilon }_0=8.85\times 10^{-12}$ C${}^2$ N${}^{-1}$ m${}^{-2}$.

Worked example. Charges $q_1=2\mu$C and $q_2=-3\mu$C are $0.3$ m apart:

$$F=\frac{9\times 10^9\times (2\times 10^{-6})(3\times 10^{-6})}{(0.3{)}^2}=\frac{0.054}{0.09}=0.6\text{ N}$$

The force is attractive, since the charges are unlike.

An everyday example. A dupatta clinging to a woollen sweater on a dry winter day is held by the Coulomb attraction between opposite charges produced by rubbing.

The substance. Coulomb's law applies to point charges at rest — for extended bodies, it must be applied to small pieces and the results added.

By the superposition principle, the force on a charge due to several charges is the vector sum of the forces each charge would exert on it alone, unaffected by the others.

$${\vec{F}}_1={\vec{F}}_{12}+{\vec{F}}_{13}+\cdots +{\vec{F}}_{1n}$$

Worked example 1 — charges in a line. $q_A=+4\mu$C is at $x=0$ and $q_B=+1\mu$C at $x=0.3$ m. Find the force on $q=+1\mu$C at $x=0.1$ m.

$$F_A=\frac{9\times 10^9\times 4\times 10^{-6}\times 10^{-6}}{(0.1{)}^2}=3.6\text{ N, towards }+x$$

$$F_B=\frac{9\times 10^9\times 10^{-6}\times 10^{-6}}{(0.2{)}^2}=0.225\text{ N, towards }-x$$

$$F_{net}=3.6-0.225=3.375\text{ N along }+x$$

Worked example 2 — zero net force. Between the charges, the forces balance where $\frac{4}{x^2}=\frac{1}{(0.3-x{)}^2}$, so $\frac{2}{x}=\frac{1}{0.3-x}$ and $x=0.2$ m.

An everyday example. A knot pulled by several ropes in a game moves according to the vector sum of all the pulls, not their simple total.

The substance. Forces add as vectors — two equal forces at right angles give $\sqrt{2}$ times one force, not double it.

**The electric field at a point is the force per unit positive test charge, $\vec{E}=\frac{\vec{F}}{q_0}$; a point charge $q$ produces $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}$, fields of several charges add as vectors, and field lines show the field's direction and, by their crowding, its strength.

Field of a point charge:**

$$\vec{E}=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}\hat{r}$$

pointing away from a positive charge and towards a negative one, in N C${}^{-1}$.

Worked example. The field $0.2$ m from a charge of $+8\mu$C:

$$E=\frac{9\times 10^9\times 8\times 10^{-6}}{(0.2{)}^2}=\frac{7.2\times 10^4}{0.04}=1.8\times 10^6{\text{ N C}}^{-1}$$

A $+2\mu$C charge placed there feels $F=qE=2\times 10^{-6}\times 1.8\times 10^6=3.6$ N.

Properties of field lines:

- They start on positive charges and end on negative charges
- They never cross, because the field has one direction at each point
- They crowd together where the field is strong
- They form no closed loops in electrostatics

An everyday example. A photocopier holds toner powder on charged regions of a drum using electric fields, before transferring it to paper.

The substance. The test charge must be tiny, so that it does not disturb the charges producing the field.