Electric charge and its basic properties (additivity

Rub a plastic comb through dry hair on a winter morning and it will lift stray strands and snap a spark against your fingertip. Nothing was "created" in that spark — a swarm of electrons, each carrying an identical, indivisible sliver of charge, simply changed address, hopping from your hair onto the comb. That one small demonstration hides the three ideas this whole chapter of electrostatics is built on: charge adds up like ordinary signed numbers, the total charge of an isolated system never changes no matter how violently you rub, touch, or spark it, and charge itself is not a smooth fluid but a coarse, grainy quantity, doled out only in whole-number multiples of one fundamental packet. Before Coulomb's law can tell you how hard two charges push or pull on each other, and before Gauss's theorem can sweep whole surfaces of charge into a single elegant statement, you need to be completely sure what the " $q$ " in every one of those formulas actually means and how it behaves. Is it a scalar or does it need vector addition like force? Can a system's total charge secretly leak away? Could a physicist someday measure a charge of $1.3e$ on some exotic particle? This article answers each question from first principles — additivity, conservation, and quantisation — and shows, with two fully worked numerical examples, exactly how CBSE examiners like to test them: touching identical conducting spheres together, and checking whether a quoted charge value is even physically allowed.

Key takeaways

Charge is a scalar, and it adds algebraically. The total charge of a system is the signed sum $q_{\text{total}} = q_1 + q_2 + \cdots + q_n$ , positive and negative contributions cancelling exactly like signed numbers, with no direction to worry about.
Charge is conserved. The total charge of an isolated system stays fixed forever; rubbing, touching, or sparking only transfers electrons between bodies, it never creates or destroys net charge.
Charge is quantised. Every free charge obeys $q = ne$ , with $n$ a positive or negative integer and $e = 1.6 \times 10^{-19}\ \text{C}$ the smallest unit any isolated body can carry.
Macroscopic "continuous" charge is an illusion of scale. A charge of a few microcoulombs corresponds to $n$ of the order $10^{13}$ electrons, so the graininess is invisible at everyday scales and charge behaves, for circuit and field calculations, as if it varied smoothly.
Three charging methods rest on these same laws. Friction, conduction (contact), and induction (no contact) all redistribute a fixed, quantised total charge between bodies — none of them manufactures new charge.
Touching identical conductors splits charge equally. By symmetry, when identical conducting spheres are brought into contact and separated, their combined charge (found by additivity) is shared out equally among them.
Key idea: electric charge always obeys three linked rules — it adds as a signed scalar, its system total is conserved, and it exists only in whole-number multiples of e — and every charging or charge-sharing problem is just careful bookkeeping under those three rules.

The core insight

Charge is a conserved, quantised scalar — it sums algebraically as $q_{\text{total}} = \sum q_i$ , the total of an isolated system never changes, and any real charge must satisfy $q = ne$ for an integer $n$ and $e = 1.6\times10^{-19}\ \text{C}$ .

Where it fits in the syllabus

This is the opening topic of the CBSE Class 12 Physics chapter "Electric Charges and Fields" (Unit I: Electrostatics), and it is squarely inside the 2025-26 prescribed scope — charge, conservation of charge, and quantisation ( $q = ne$ ) are listed explicitly, ahead of Coulomb's law and the electric field. Everything that follows in the chapter — Coulomb's law, superposition for many charges, the field of a point charge, the dipole, and Gauss's theorem — treats " $q$ " as a signed, conserved, quantised number, so this topic is the vocabulary lesson the rest of electrostatics depends on. Nothing here touches the chapter's out-of-scope items (the formal proof of Gauss's theorem, fields of non-symmetric or thick charge distributions); those come later and use, without re-deriving, the same charge properties established here.

The physical picture

Picture charge not as a continuous fluid you could pour in any amount, but as a collection of identical, countable tokens — like coins rather than water. Two elementary carriers fix the scale of every token: the electron carries $-e$ and the proton carries $+e$ , with $e = 1.6\times10^{-19}\ \text{C}$ . Any isolated object's total charge is simply the count of surplus or deficit tokens times $e$ :

q = n e, \qquad n = 0, \pm1, \pm2, \pm3, \ldots

where: $q$ is the net charge on the body (in coulombs), $n$ is a (positive, negative, or zero) integer counting the excess electrons ( $n<0$ ) or deficit of electrons ( $n>0$ ), and $e$ is the elementary charge, a fixed constant of nature. This is quantisation — you can never, even in principle, isolate a free charge of $0.5\,e$ or $1.37\,e$ ; nature's charge "coins" do not come in fractions (quarks inside protons and neutrons carry fractional charge, but they are never found alone, so every observable free charge remains an integer multiple of $e$ ). When several charged bodies are grouped together as one system, the total charge is found by ordinary signed addition — this is additivity, and it is what makes charge a scalar rather than a vector:

q_{\text{total}} = q_1 + q_2 + q_3 + \cdots + q_n

where: each $q_i$ carries its own sign, so a $+5\ \mu\text{C}$ charge and a $-3\ \mu\text{C}$ charge in the same system contribute a net $+2\ \mu\text{C}$ , not $8\ \mu\text{C}$ — there is no direction or angle to account for, unlike combining forces or velocities. The third law, conservation, is a statement about time rather than about combination: however the charges inside an isolated system move, collide, transfer, or redistribute, the algebraic sum $q_{\text{total}}$ computed above stays exactly fixed. Nothing in electrostatics, chemistry, or ordinary charging processes creates a lone $+$ charge or a lone $-$ charge from nothing — charge only ever moves from one place to another, most commonly as a flow of electrons (the far lighter, far more mobile carrier compared with the nucleus-bound protons).

Figure

Two neutral bodies before rubbing shown side by side with equal small circles of plus and minus charge, and after rubbing the same two bodies shown again with electrons having crossed from the left body to the right body, the left body now net positive and the right body now net negative while the combined total charge label reads zero both before and after

Charge transferred by rubbing, tied to the number of electrons

Derivation ALinking a measured mass change to the number of electrons transferred

The goal is to connect two things a student might think are unrelated: a tiny, in-principle measurable mass change of a rubbed body, and the number of electrons it has gained or lost, using only the quantisation law $q = ne$ and the electron's known mass. This matters because it turns "charge is quantised" from an abstract slogan into an arithmetic tool you can use on any charging-by-friction question, and it shows explicitly why conservation of charge and conservation of the number of electrons are really the same statement viewed two ways.

Step 1. Suppose a rod loses a certain number of electrons, $n$ , to a duster during rubbing. By quantisation, the charge it has gained (having lost negative charge, it becomes net positive) is $q = n e$ , where $e = 1.6\times10^{-19}\ \text{C}$ is the charge magnitude of one electron.

Step 2. By conservation of charge for the isolated (rod + duster) system, whatever positive charge $q$ appears on the rod must be exactly matched by an equal negative charge $-q$ appearing on the duster — the same $n$ electrons that left the rod are now sitting on the duster. So $q_{\text{rod}} + q_{\text{duster}} = ne + (-ne) = 0$ at every instant of the rubbing, not just at the end.

Step 3. Each transferred electron also carries a tiny mass $m_e = 9.1\times10^{-31}\ \text{kg}$ , so the rod's mass changes by $\Delta m = n\,m_e$ (a decrease, since it lost electrons). Combining with Step 1, $n = q/e$ , so $\Delta m = \dfrac{q}{e}\,m_e = q\left(\dfrac{m_e}{e}\right)$ .

Step 4. Evaluate the constant $m_e/e = \dfrac{9.1\times10^{-31}}{1.6\times10^{-19}} = 5.7\times10^{-12}\ \text{kg/C}$ . So even a rod charged to a hefty $q = 1\ \mu\text{C} = 10^{-6}\ \text{C}$ only loses $\Delta m = (10^{-6})(5.7\times10^{-12}) \approx 5.7\times10^{-18}\ \text{kg}$ — utterly unmeasurable on any ordinary balance, which is exactly why charging by friction "feels" like it creates something from nothing, when in fact it is a bookkeeping-perfect transfer of a minuscule number of electrons ( $n = q/e \approx 6.25\times10^{12}$ electrons for that same $1\ \mu\text{C}$ ). Check: the same $n$ appears in both the charge balance ( $q_{\text{rod}} = -q_{\text{duster}}$ ) and the mass balance ( $\Delta m_{\text{rod}} = -\Delta m_{\text{duster}}$ ), confirming conservation of charge and conservation of electron number are one and the same fact expressed in different units.

Worked examples

Board level — three touching spheres, redistributing a mixed charge

Question: Three small identical conducting spheres carry charges $+3\ \mu\text{C}$ , $-5\ \mu\text{C}$ , and $+2\ \mu\text{C}$ . They are brought together so all three touch simultaneously, allowed to reach a common charge, and then separated. Find the net charge of the system and the final charge on each sphere.

Step 1. Apply additivity to find the total charge before contact: $q_{\text{total}} = (+3) + (-5) + (+2) = 0\ \mu\text{C}$ .

Step 2. Apply conservation of charge: touching does not create or destroy charge, so the combined total immediately after separation must still equal $0\ \mu\text{C}$ , exactly as before contact.

Step 3. Because the three spheres are identical (same size, same material), symmetry forces the shared charge to divide equally among them once they are in mutual contact, regardless of how it was distributed originally: each final charge is $q_{\text{total}}/3$ .

Step 4. Substitute the value from Step 2: each sphere ends up with $0\ \mu\text{C}/3 = 0\ \mu\text{C}$ .

Answer: the net charge of the system is $\mathbf{0}$ , and after separation every sphere is exactly neutral — a compact illustration of how additivity and conservation together, not any new physical law, fully determine the outcome.

Competitive level — checking whether two quoted charges are physically allowed

Question: A charged oil drop in a Millikan-type setup is measured to carry $q_1 = 8.0\times10^{-18}\ \text{C}$ . A second drop is measured to carry $q_2 = 4.5\times10^{-19}\ \text{C}$ . Determine which value (if either) is a physically possible free charge, and, for the allowed one, find the number of excess or deficit electrons.

Step 1. By quantisation, any physically real free charge must satisfy $q = ne$ for some integer $n$ , with $e = 1.6\times10^{-19}\ \text{C}$ .

Step 2. Test $q_1$ : $n_1 = \dfrac{q_1}{e} = \dfrac{8.0\times10^{-18}}{1.6\times10^{-19}} = 50.0$ . This is an exact integer, so $q_1$ is allowed, corresponding to a body with 50 excess or deficit electrons.

Step 3. Test $q_2$ : $n_2 = \dfrac{q_2}{e} = \dfrac{4.5\times10^{-19}}{1.6\times10^{-19}} = 2.8125$ . This is not an integer, so no arrangement of whole electrons can produce exactly this charge — $q_2$ cannot occur on an isolated body.

Step 4. State the physical reasoning an examiner is checking: quantisation is not a rounding convenience, it is exact, so any non-integer value of $q/e$ , however close to a whole number, marks the quoted reading as either a measurement error or a fictitious value in the problem.

Answer: $q_1 = 8.0\times10^{-18}\ \text{C}$ is physically possible, with $\mathbf{n_1 = 50}$ electrons of imbalance; $q_2 = 4.5\times10^{-19}\ \text{C}$ is not possible, since $q_2/e = 2.8125$ is not an integer. Timing note: this whole check should take under ninety seconds in an exam — divide the quoted charge by $1.6\times10^{-19}\ \text{C}$ and inspect the decimal; a non-integer result is an instant, decisive "not allowed" without needing any further physics.

Common traps

Students lose marks on this topic in remarkably predictable ways, almost always from treating one of the three properties too loosely.

Adding magnitudes instead of signed values. A common slip is writing $q_{\text{total}} = 3 + 5 + 2 = 10\ \mu\text{C}$ for the worked example above, forgetting that additivity is an algebraic sum — the sign of every charge must be carried through the addition, and opposite charges genuinely cancel.
Treating friction as charge creation. Describing a rubbed rod as having "gained charge" without saying where that charge came from suggests charge appeared from nowhere; always phrase it as electron transfer between the two bodies, with the system total unchanged.
Assuming any decimal multiple of $e$ is "close enough." Quantisation is exact, not approximate — $q/e$ must land on a whole number, and a value like $2.8$ or $50.3$ is simply not permitted for a free charge, not merely "unlikely."
Confusing conduction and induction. Conduction charging requires physical contact and always leaves both bodies with charge of the same sign as the originally charged body (shared out by contact); induction needs no contact, separates charge on a neutral conductor (near side opposite sign, far side same sign as the inducing charge), and can leave a net charge opposite to the inducing body only if the conductor is earthed and then disconnected while the inducing charge is still nearby.
Forgetting the "identical bodies" condition for equal sharing. The rule "touching conductors share charge equally" only holds when the bodies are identical in size and shape; unequal or differently shaped conductors share charge in a way that depends on their geometry, which is beyond what additivity and conservation alone can predict.

Common trapQuantisation is exact, not a rounding rule — and charge conservation applies to the whole isolated system, not to each body separately

The single biggest misconception is expecting charge to behave like a continuous quantity that can be "approximately" any value, and treating conservation as if each individual body's charge must stay fixed. Neither is true. Quantisation is a hard constraint: $q/e$ must be a whole number, full stop, with no tolerance for "almost integers" — a measured or quoted charge that fails this test is not a real free charge, however small the discrepancy looks. Conservation, meanwhile, is a statement about the isolated system as a whole: an individual body's own charge can and does change freely (that is exactly what charging by friction, conduction, or induction does), as long as whatever charge it gains is lost by another body in the system, so the grand total stays fixed. Mixing these two ideas up — expecting each body to keep its own charge fixed, or expecting the total to drift by a "small" non-quantised amount — is the fastest way to lose marks on charge-transfer numericals.

You should be able to…

State and correctly apply all three basic properties of charge: additivity, conservation, and quantisation.
Compute the total charge of a system as a signed (algebraic) sum, never as a sum of magnitudes.
Test whether a quoted charge value is physically allowed by checking that $q/e$ is an integer, and find that integer $n$ .
Predict the final charge on identical conducting spheres after they are touched together and separated.
Explain, in terms of electron transfer, why friction charging leaves two bodies oppositely charged while their combined total stays zero.
Distinguish friction, conduction, and induction as charging methods, including which one can leave a net charge on an initially neutral conductor without contact.

Ask the tutor: Want to work through a trickier induction-charging scenario, or check your own worked answer for a multi-sphere contact problem step by step? Ask the AI tutor.

If charge is quantised, why do we treat it as continuous almost everywhere else in physics, like in circuit equations?

Because the elementary charge $e = 1.6\times10^{-19}\ \text{C}$ is so tiny compared with everyday charges that the graininess is completely invisible at any scale you could measure with ordinary lab equipment. A current of just one microampere already involves roughly six trillion electrons flowing past a point every second; treating that flow as a smooth continuous quantity introduces no meaningful error, in exactly the way treating sand as a continuous fluid works fine for building a sandcastle even though it is made of discrete grains. Quantisation only becomes something you have to think about explicitly when you are dealing with a handful of elementary charges at a time, such as a single charged oil drop or a single electron hopping in a nanoscale device.

Does conservation of charge mean a single charged ball can never lose its charge?

No — an individual body's own charge is free to change; what is fixed is the total charge of every body involved, taken together as one isolated system. A charged ball left in humid air will slowly lose its charge as it neutralises through contact with moisture and dust in the environment, but that charge has not vanished — it has been transferred to the air, the moisture, and the surfaces the ball touches, so the true "system" (ball plus everything it exchanges charge with) still conserves the total. In a truly isolated system, with no leakage path to the outside, an individual body's charge would indeed stay fixed forever unless something inside the system redistributes it — but real-world charged objects are essentially never perfectly isolated from their surroundings.

Why does touching two identical spheres split the charge equally, but touching two different-sized spheres does not?

Equal sharing between identical spheres follows purely from symmetry: since the two spheres are indistinguishable in size, shape and material, there is no physical reason for the shared charge to prefer one over the other, so additivity and conservation alone (total charge divided two ways, with no asymmetry to break the tie) force an equal split. Different-sized conducting spheres, however, reach a common potential when touched rather than a common charge, and potential depends on geometry (a larger sphere can hold more charge at the same potential) — working out exactly how the charge splits then needs the electrostatic potential relations from later in this chapter's sibling topic, not just additivity and conservation. That is why CBSE numericals on this exact topic almost always specify "identical spheres" — it keeps the problem solvable with only the three basic properties covered here.

Is it possible for the number n in q equals n e to be negative?

Yes, and it should be — a negative $n$ simply describes a body with a net surplus of electrons rather than a deficit. By convention here, $n$ counts electrons lost from a neutral state as positive and electrons gained as negative (or you may see the opposite sign convention in a given textbook, as long as it is applied consistently): a rod that has lost 50 electrons during rubbing carries $q = +50e$ , while the silk it rubbed against, having gained those same 50 electrons, carries $q = -50e$ . Whichever sign convention is used, the magnitude of $n$ tells you the number of elementary charges involved, and the sign simply tells you whether the body is electron-deficient (positive) or electron-rich (negative).