Electric charge: properties, quantization and conservation

Rub a balloon on your hair and it sticks to a wall — a party trick that is really the whole of electrostatics in miniature. Behind it lie two of the deepest bookkeeping rules in physics: charge is never created or destroyed, only moved, and whatever charge a body ends up with is always a whole-number multiple of one indivisible packet, the electron's charge ee. This section builds the vocabulary — positive and negative charge, conductors and insulators, the three ways to charge a body — and the two governing laws, conservation and quantization, that every later electrostatics calculation (Coulomb's law, fields, potential) silently assumes. Everything here is algebra: no calculus, just counting electrons and adding up totals.

The core insight

Electric charge on any isolated body is conserved in total and quantized as q=neq = ne, because it is carried only by whole electrons hopping from one body to another, never created, destroyed, or split.

The core idea

Two kinds, one rule. There are exactly two kinds of charge, positive and negative; like charges repel, unlike charges attract. Charge is an intrinsic property of the electron and proton — the electron carries −e-e, the proton +e+e, and a neutral atom has equal numbers of each. Objects charge up only by transferring electrons; protons are locked inside the nucleus and never move in ordinary charging.

Conservation of charge. The total charge of an isolated system never changes — it can only be transferred between the bodies that make it up. Rub a glass rod with silk and electrons hop from rod to silk: the rod ends up positive, the silk equally negative, and the sum over rod-plus-silk stays exactly zero throughout.

Quantization of charge. Because every electron carries the same fixed charge −e-e, any charge qq you could ever measure on an isolated body must be a whole-number count of electrons transferred in or out:

q=ne,n=0,±1,±2,±3,…q = ne, \qquad n = 0, \pm1, \pm2, \pm3, \ldots

where e=1.6×10−19 Ce = 1.6\times10^{-19}\ \text{C} is the elementary charge and nn is an integer (never a fraction). For lab-scale charges (n∼1010n \sim 10^{10} or more) the graininess is far too fine to detect, so charge looks continuous — like a beach that looks smooth even though it is made of individual grains of sand.

Conductors and insulators. In conductors (metals), some electrons are only loosely bound and roam freely through the material; in insulators (glass, rubber, plastic) all electrons stay put, so charge placed on one spot stays there.

Three ways to charge a body. Friction transfers electrons between two different neutral materials on contact and rubbing. Conduction is direct touch between a charged and a neutral conductor: charge spreads until both share a common state, and the neutral body ends up with the same sign as the charged one. Induction needs no contact: bringing a charged rod near a neutral conductor pushes its free electrons around, so the near face gets an induced charge opposite the rod's and the far face the same sign; grounding the far face and then removing first the ground and then the rod leaves the conductor charged opposite to the rod — while the rod itself loses nothing.

Figure
Comparing conduction and induction charging of a neutral metal sphere by a positively charged rod
Conduction gives the same sign; induction gives the opposite sign Conduction — touching + rod neutral touch, then separate sphere ends up + (same sign) Induction — no touch, then ground + rod − + ground far side remove ground, then rod sphere ends up − (opposite sign)
Derivation AWhy charge must come in whole multiples of e

The goal is to show, from nothing more than the idea that electrons are the only mobile charge carriers in ordinary matter and every electron carries exactly the same charge −e-e, that the net charge picked up by any body during friction, conduction or induction is forced to be an integer multiple of ee — never a fraction, and always consistent with the total staying fixed for an isolated system.

Step 1. Start from what actually moves. In friction, conduction and induction alike, the only particles that transfer between bodies are electrons; protons are bound inside nuclei and never hop across. So any change in a body's charge is built up one electron at a time.

Step 2. Assign the elementary unit. Each electron that leaves a body removes exactly −e-e from it, which is the same as adding +e+e to its charge; each electron that arrives adds exactly −e-e. There is no intermediate amount — an electron cannot be half-transferred.

Step 3. Add up whole electrons. If a body starts neutral and ends up having lost nn electrons (equivalently gained nn units of positive charge), its final charge is the sum of nn identical contributions of size +e+e: q=e+e+⋯+eq = e + e + \cdots + e (nn times) =ne= ne. Because nn counts discrete electrons, it must be a whole number — positive if electrons were removed, negative if electrons were added, zero if none moved.

Step 4. Reconcile with conservation. Whatever charge one body gains as +ne+ne, the body it exchanged electrons with loses exactly −ne-ne, so the sum over the two bodies is unchanged — conservation and quantization are two faces of the same electron-counting process, not independent assumptions. Check: this reasoning never invokes the size of ee itself, only that transfers happen in whole electrons — so the rule q=neq = ne holds for any charging method (friction, conduction, induction) and for any pair of bodies, isolated or in contact.

Worked example

Sphere A carries qA=+12 nCq_A = +12\ \text{nC} and identical sphere B carries qB=−4 nCq_B = -4\ \text{nC}. The spheres are touched together and separated. Find the final charge on each sphere, and the number of electrons that crossed between them to reach that state.

Step 1. Find the conserved total before contact: qtotal=qA+qB=12 nC+(−4 nC)=8 nCq_{\text{total}} = q_A + q_B = 12\ \text{nC} + (-4\ \text{nC}) = 8\ \text{nC}. By conservation of charge this total is unchanged by the contact.

Step 2. Because the spheres are identical conductors, symmetry forces the total to split equally on separation: qA′=qB′=qtotal2=4 nCq_A' = q_B' = \dfrac{q_{\text{total}}}{2} = 4\ \text{nC} each.

Step 3. Track the electron transfer using quantization. Sphere B went from −4 nC-4\ \text{nC} to +4 nC+4\ \text{nC}, a change of +8 nC+8\ \text{nC}, which means it lost electrons equal to Δn=8×10−91.6×10−19=5×1010\Delta n = \dfrac{8\times10^{-9}}{1.6\times10^{-19}} = 5\times10^{10} electrons (these moved onto A, which correspondingly gained charge and lost the matching number of excess electrons it started with).

Both spheres end up at +4 nC+4\ \text{nC}, reached by a transfer of 5×10105\times10^{10} electrons from B to A — an astronomically large but still exactly integral count, consistent with q=neq = ne.

Common traps

  • Thinking friction creates charge — it only transfers electrons between two initially neutral bodies, so the net charge of the pair stays zero.
  • Mixing up conduction and induction: conduction needs contact and gives the same sign as the charging body; induction needs no contact and (after proper grounding) gives the opposite sign, leaving the charging body's own charge unchanged.
  • Assuming quantization means everyday charges "look grainy" — in reality nn is enormous (often 101010^{10} or more) for lab-scale charges, so discreteness is never seen directly.
  • Forgetting that only electrons move in ordinary charging; protons stay locked in the nucleus, so a positively charged body has lost electrons, not gained protons.

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If charge is quantized, why do we treat it as continuous in Coulomb's law and field formulas?

Because the elementary charge e=1.6×10−19 Ce = 1.6\times10^{-19}\ \text{C} is so tiny that any lab-scale charge (microcoulombs and up) corresponds to nn in the billions or more. At that scale, treating nn as a continuous variable introduces negligible error, exactly like treating a beach as a smooth surface even though it is made of discrete grains of sand. Quantization only matters conceptually — it tells you which charge values are physically allowed — not in everyday force or field calculations.

Does induction charging violate conservation of charge, since the neutral sphere ends up charged out of nowhere?

No — look at the whole system, not just the sphere. The charging rod's own charge never changes throughout the process; what happens is that the sphere's own electrons (which it already had, in equal number to its protons) redistribute internally, and some are removed to the ground during the grounding step. The charge that leaves through the ground wire is real charge conserved with the ground (usually the Earth, effectively infinite), not created from nothing.

Why can't protons be transferred to charge an object, the way electrons are?

Protons sit inside the atomic nucleus, held there by the strong nuclear force, which is far stronger than anything at play in rubbing, touching or bringing objects near each other. Electrons, in contrast, occupy the outer regions of atoms and in conductors are only weakly bound, so ordinary mechanical or electrical contact easily dislodges them. This is why every charging method in this syllabus — friction, conduction, induction — is described purely in terms of electron transfer, never proton transfer.

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