Physics · Electricity and magnetism · Grade 7

Electrification, Coulomb's law

Why does hair follow a balloon after it has been rubbed on it?

The situation

On a dry winter day, if you rub a balloon on your hair, your hair stands on end and follows the balloon, while the balloon sticks to the wall. Two foil-covered balls hung on silk threads, once charged, either fly apart or are drawn together. What does the force between charges depend on — and how can the charge of a ball be measured from its angle of deflection?

Topic

Electrification, Coulomb's law. Balloon and hair: static electricity

Key ideas

Coulomb's law
Electrification
Conservation of charge

Formulas

F = k \frac{| q_{1} q_{2} |}{r^{2}}, k = \frac{1}{4 π ε_{0}} \approx 8.99 \cdot 10^{9} \frac{{N·m}^{2}}{C^{2}}

Coulomb's law: the force is directly proportional to the product of the charges and inversely proportional to the square of the distance. $ε_{0} \approx 8.85 \cdot 10^{- 12}$ F/m is the electric constant.

\tan α = \frac{F}{m g}, r = d + 2 L \sin α

Equilibrium of a hanging ball (the sum of $m g$ , $T$ and $F$ is zero) and the distance between the balls: $d$ is the distance between the suspension points, $L$ is the distance from the suspension point to the centre of the ball.

W = k \frac{q_{1} q_{2}}{r}

Interaction potential energy of two point charges (zero at $r \to \infty$ ). For like charges $W > 0$ .

q = r \sqrt{\frac{m g \tan α}{k}}

Finding the charge from the angle of deflection for equally charged balls ( $q_{1} = q_{2} = q$ ).

q_{1}^{'} = q_{2}^{'} = \frac{q_{1} + q_{2}}{2}

When identical conducting balls touch, the charge is shared equally — the total charge is conserved.

Theory

Electrification — the transfer of electrons

When two bodies are rubbed together, electrons pass from the first to the second. In the triboelectric series hair comes before the rubber balloon: hair gives up electrons and becomes positively charged, the balloon gains electrons and becomes negatively charged. Charge does not appear from nothing: the sum of the charges of balloon and hair is always zero — check this in “Balloon and hair” mode. Charge is discrete: any charge is a multiple of the elementary charge, $q = N e$ , $e = 1.602 \cdot 10^{- 19}$ C (the electron has charge $- e$ , the proton $+ e$ ). A balloon rubbed once (≈ −175 nC) has about $1.1 \cdot 10^{12}$ excess electrons — $N$ is shown in the measurements.

Coulomb's law

Like charges repel, unlike charges attract. The force is proportional to the product $q_{1} q_{2}$ and inversely proportional to the square of the distance: if the distance is halved, the force increases 4 times. On the “Force — distance” graph the measured points always lie on the theoretical $1 / r^{2}$ curve. By Newton's third law the force on both balls is equal, so the angles of deflection are the same even if $q_{1} \neq q_{2}$ .

Why does the ball come to rest at a certain angle?

Three forces act on the ball: $m g$ downward, $F$ horizontal, and the string tension $T$ along the string. At equilibrium $\tan α = F / m g$ . But $F$ depends on the distance, and the distance on the angle — so the simulation moves the balls like real pendulums (with air resistance) and they settle into equilibrium on their own. At $α = 45 °$ , $F = m g$ .

Electroscope, electrometer and electrophorus

When a charged balloon is brought near an electroscope, the electrons in the conductor are redistributed (electrostatic induction): charge of the same sign as the balloon collects on the leaves and they open. If the balloon touches it, part of the charge passes across and the leaves remain open even after the balloon is moved away. An electrometer is an electroscope with a metal case and a scale: charge or potential is measured from the deflection of the pointer. If, while the balloon is near, you earth the electroscope, charge of the same sign flows to earth and the opposite charge remains — this is charging by induction. An electrophorus (a metal disc on a charged dielectric plate) and an electrostatic machine (rotating discs) separate charge again and again in this way and produce a spark several centimetres long.

Units of charge, the electric constant and conservation

In SI the unit of charge is the coulomb: 1 C = 1 A·s, that is, the charge passing in 1 s at a current of 1 A. This is very large: the charge on a ball is measured in nanocoulombs (1 nC = $10^{- 9}$ C). In Coulomb's law $k = 1 / (4 π ε_{0})$ , where $ε_{0} \approx 8.85 \cdot 10^{- 12}$ F/m is the electric constant. In the CGS system $F = q_{1} q_{2} / r^{2}$ ( $k = 1$ ) is written and the unit of charge is the statcoulomb (CGSE): 1 CGSE ≈ $3.34 \cdot 10^{- 10}$ C. The local form of the conservation law is the continuity equation $\partial ρ / \partial t + \nabla \cdot \vec{j} = 0$ : the charge in a volume changes only because of current flowing in or out through its surface.

Potential energy and the model

In bringing charges closer together, work is done against the Coulomb force and is stored as potential energy $W = k q_{1} q_{2} / r$ . For like charges $W > 0$ and it increases as $r$ decreases; for unlike charges $W < 0$ . The work done by the Coulomb force, $A = W_{1} - W_{2}$ , does not depend on the path. Watch $W$ in the table of measurements. Model: the balls are treated as point charges ( $r ≫ R$ ), and their polarisation is ignored; the models of hair and electroscope are approximate. In dry air a ball of radius $R = 1.3$ cm cannot hold more than ≈ 55 nC — the field at the surface reaches 3 MV/m and the air breaks down.

Did you know?

Coulomb's law was established in 1785 by the French physicist Charles-Augustin de Coulomb, using a torsion balance that he built himself.
Dry air breaks down in a field of about 3 MV/m — a spark appears. That is why a ball of radius 1 cm cannot hold much more than about 30 nC.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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