Charge, Conservation of Charge, Current and Voltage
Part of B.5 Current and Circuits
· Unit B — IB Physics 2023
1 Lesson 1 of 7 · B.5 Current and Circuits
Charge, Conservation of Charge, Current and Voltage
Every circuit you will ever analyse rests on two conserved quantities: charge and energy. This lesson builds current and potential difference directly from those two conservation laws, so the equations aren't rules to memorise — they're bookkeeping.
🎯 Hook – the most dangerous experiment in physics history
In 1752, Benjamin Franklin flew a kite into a thunderstorm to show that lightning is an electrical discharge. He wasn't being reckless for its own sake — nobody yet knew what "electricity" really was, and the only way to find out was to go and get some. He survived, but several people who repeated the experiment did not.
Fig 1 — Franklin's kite experiment. The kite string, made damp by the storm, conducted charge from the cloud down to a key tied near the ground.
💡 Diagnostic question (think — then read on) — a spark jumps from the key to Franklin's knuckle. What is actually moving? Charged particles — electrons — flowing from one place to another. That flow, however brief, is an electric current. The lightning bolt itself is nothing more or less than a very large, very fast current.
⚡ Electric charge
Electric charge is a fundamental property of some subatomic particles, responsible for the forces between them. There are two kinds of electric force — attractive and repulsive — so we need two kinds of charge to explain them: positive charge and negative charge. Calling them "positive" and "negative" carries no deeper meaning beyond labelling two different types of the same thing; they are often described as opposite charges.
✓ Charges of opposite sign attract each other.
⚠️ Charges of the same sign repel each other.
Charge is measured in coulombs, C. One coulomb is a relatively large amount of charge, so microcoulombs (\(1\ \mu\text{C} = 10^{-6}\ \text{C}\)) and nanocoulombs (\(1\ \text{nC} = 10^{-9}\ \text{C}\)) are common in practice.
Fig 2 — A simple atom model (not to scale). Protons and neutrons sit in the small central nucleus; electrons occupy the space around it.
All protons carry a positive charge of \(+1.60\times10^{-19}\ \text{C}\) and all electrons carry a negative charge of \(-1.60\times10^{-19}\ \text{C}\). Neutrons carry no charge at all — they are neutral.
Elementary charge, e
$$e = 1.60\times10^{-19}\ \text{C}$$
Since 2019, the elementary charge has been defined to be exactly \(1.602176634\times10^{-19}\) C — it is no longer just a measured value.
🤔 TOK — should all scientific knowledge be pursued regardless of risk? Franklin's kite experiment expanded scientific knowledge but endangered lives; some who repeated it were killed. A notorious 1903 demonstration used electricity to kill a "rogue" circus elephant, Topsy, raising separate questions about the ethics of experimenting on animals. Opinions on where the line should be drawn vary widely and are not settled by physics itself.
🔢 Charge is quantized — and conserved
A continuous variable can, in principle, take any value; a discrete variable can only take certain values — like a carton of eggs, which can hold 6 eggs or 10, but never 6.5. A quantity restricted to certain discrete values is described as quantized.
Any quantity of charge consists of a whole number of charged particles, each \(\pm1.60\times10^{-19}\) C — intermediate values such as \(4.00\times10^{-19}\) C are impossible (ignoring sub-nuclear quarks, which are outside this course). We say charge is quantized. Charge is generally given the symbol \(q\) (occasionally \(Q\), though \(Q\) is also used for thermal energy, so it can be ambiguous).
Number of electrons in 1 C
$$\frac{1}{1.60\times10^{-19}} = 6.24\times10^{18}\ \text{electrons}$$
📖 Law of conservation of charge — the total charge in an isolated system remains constant. This is one of the few conservation rules in physics that is always true, without exception.
If one or more negatively charged electrons are removed from a neutral atom, conservation of charge tells us the remaining atom must carry an equal positive charge. The charged atom is then called an ion, and the process is ionization.
🔋 Electric current
Whenever charge flows from place to place, we call it an electric current. Electrons, protons and ions are all charged particles, but a moving charge in a current is generally called a charge carrier — and to flow, a charge carrier must be relatively "free" to move.
Some electrons in metal atoms have enough energy that they are no longer attracted to a particular metal ion — these delocalized electrons (or free electrons) are why electric current flows through metals far more readily than through most other materials.
Fig 3 — Free electrons drift through a metal wire towards the positive battery terminal, superimposed on their much faster random thermal motion. Point P marks where we count the flow of charge.
With no current flowing, free electrons move around randomly at high speed, rather like molecules in a gas. When they form a current, a much slower drift velocity in the direction of the current is added on top of that random motion — far too slow and chaotic to be shown accurately in a single diagram.
Electric current · SI unit ampere, A
$$I = \frac{\Delta q}{\Delta t}$$
Current, \(I\), is the amount of charge passing a point (such as P above) per unit time. The SI unit is the ampere (amp), A — milliamps (mA) and microamps (μA) are also common. It is one of the seven base SI units, defined as the current in which 1 C of charge (\(6.24\times10^{18}\) electrons) passes a point in one second. (Before 2019 it was defined instead via the magnetic force between two long parallel current-carrying wires — not needed here.)
A current that only flows in one direction is a direct current (dc); one that continuously reverses is an alternating current (ac).
✓ Top tip: dc is generally more useful for electronics, but ac is used to transmit electrical energy around the world, because it is far more easily transformed to the high voltages that reduce energy losses in the wires.
🔌 Circuits, conventional current, and ideal meters
For current to flow continuously, components and wires must form a complete, closed loop — an electrical circuit — and it must include a battery or other power source. A single electric cell is often loosely called a "battery," though strictly a battery is a combination of two or more cells.
Fig 4 — A simple circuit, drawn conventionally: cell, resistor, switch and ammeter in one loop.
The positive terminal of a battery attracts free electrons from the circuit and the negative terminal repels them, so electrons actually drift around a circuit like Fig 4 from the negative terminal, through the components, to the positive terminal. But for historical reasons — chosen long before electrons were even discovered — we use the opposite convention:
⚠️ Conventional current is always shown flowing from positive to negative around a circuit — opposite to the actual direction of electron flow.
✓ Top tip: it helps to think of current as flowing everywhere in the circuit at the same time, rather than starting or finishing at the battery. (In reality, an electric field sets electrons in motion around the whole circuit at a speed close to the speed of light.)
Ammeters measure current, and must be connected in series so that all the current to be measured flows through them. An ideal ammeter has zero resistance, so it doesn't reduce the current it's measuring — an electronic current sensor instead detects the magnetic field around the current.
🌍 Nature of science — shared symbols across cultures. Standard circuit symbols and unit abbreviations let scientists and engineers from different countries communicate without translation errors. In 1983, an aircraft flying over Canada ran out of fuel mid-flight because of confusion between two different units used for fuel volume — a reminder of how costly mismatched units and symbols can be.
🔺 Potential difference (voltage)
"Voltage" is the everyday word; the correct term is potential difference (p.d.), symbol \(V\) — the same letter used for its unit, the volt.
Potential difference · SI unit volt, V
$$V = \frac{W}{q}$$
$$1\ \text{V} = 1\ \text{J}\ \text{C}^{-1}$$
The p.d. across two points is the work done per unit charge moving between them. A 12 V battery, for instance, transfers 12 joules of energy to every coulomb of charge that flows through it.
Voltmeters are connected in parallel across whatever they're checking. An ideal voltmeter has infinite resistance, so no current flows through it and it doesn't affect the p.d. it's measuring.
Fig 5 — Voltmeters are always connected in parallel. If \(V_{\text{s}}=12\) V and \(V_1=3\) V, \(V_2=4\) V, then \(V_3\) must be \(12-3-4=5\) V, and \(V_{12}\) (across components 1 and 2 together) reads \(3+4=7\) V.
✓ In any series loop, the p.d.s across the individual components add up to the supply p.d. — because the energy transferred into the circuit must equal the energy used by the components in it (assuming no energy is lost in the wires).
Instantaneous current = gradient of the tangent at that point
Current–time (steady current)
Time · Current, \(I\)
Horizontal line
Area under the line = charge that flowed, \(\Delta q = I\Delta t\)
Current–time (varying current)
Time · Current, \(I\)
Curve of any shape
Area under the curve = total charge, \(\Delta q\), found by counting squares or an equal-area rectangle
Fig 6 — A steady current: \(q\) against \(t\) is a straight line through the origin, and its gradient is \(I\).
Fig 7 — A varying current: the shaded area under an \(I\)–\(t\) graph equals the total charge that has flowed.
✍️ Worked example — potentials around a series circuit
A 12 V battery drives current around a series circuit through three components, P, Q and R. Taking the negative terminal of the battery as 0 V, the potential at point A (just after the battery) is 12 V; it then falls to 9 V at point B, and to 4 V at point C, before returning to 0 V. Find the p.d. across each of P, Q and R.
1
Identify: a charge moves from higher to lower electrical potential energy as it passes through each component — the same idea as a mass falling from higher to lower gravitational potential energy. The p.d. across a component is the drop in potential across it.
2
Data: potential at A = 12 V, at B = 9 V, at C = 4 V, at the negative terminal = 0 V. P lies between A and B, Q between B and C, R between C and the negative terminal.
3
Equation: p.d. across a component = (potential before it) − (potential after it). \(V_{\text{P}} = 12 - 9\) \(V_{\text{Q}} = 9 - 4\) \(V_{\text{R}} = 4 - 0\)
4
Answer: \(V_{\text{P}}=3\ \text{V}\), \(V_{\text{Q}}=5\ \text{V}\), \(V_{\text{R}}=4\ \text{V}\). Check: \(3+5+4=12\ \text{V}\) — exactly the battery's p.d., as it must be.
Fig 8 — Potential falls in steps around the loop as charge passes through each component; the p.d. across a component is the size of the step.
✍️ Worked example — charge and electrons through a lamp (B5.1)
The current through an LED desk lamp is 50 mA. (a) Calculate the charge that flows through the lamp in 1.0 minute. (b) How many electrons flow through the lamp every minute?
1
Identify: a steady current, so \(I=\Delta q/\Delta t\) rearranges directly for \(\Delta q\); then divide by \(e\) for the number of electrons.
Electricity is supplied to homes at around 110 V or 230 V depending on the country. In the lab, low voltages keep experiments safe: batteries typically supply 9 V or less, most commonly 1.5 V, while adjustable low-voltage (LT) supplies connected to the mains give around 12–15 V and must be checked regularly — including their earthing.
✓ Examiner tip: batteries also carry an environmental cost, since large numbers eventually need safe disposal — a genuine trade-off against the safety benefit of low-voltage supplies.
🚀 HL Extension — instantaneous vs average current
\(I=\Delta q/\Delta t\) gives the average current over an interval. When current varies, the instantaneous current at one instant is the gradient of the \(q\)–\(t\) graph at that point, \(I = \dfrac{\text{d}q}{\text{d}t}\) — exactly the calculus idea behind reading a curved graph's gradient at a single point rather than between two points.
Challenge: a current rises steadily from 0 to 2.0 A over 4.0 s. What charge flows? (Answer: the \(I\)-\(t\) graph is a triangle of base 4.0 s and height 2.0 A, so \(\Delta q = \tfrac12\times4.0\times2.0 = 4.0\ \text{C}\) — the average current, 1.0 A, times the time also gives 4.0 C.)
🔬 Tools, observer effect and a linking question
🌍 Nature of science — the observer effect. Any measurement can slightly disturb the thing being measured. A car-tyre pressure gauge lets a little air escape into itself, very slightly lowering the tyre pressure; many thermometers must absorb or emit thermal energy to reach equilibrium with what they're measuring; even a patient's blood pressure can rise simply because it's being measured. This is exactly why we care about ideal ammeters and voltmeters — they're the electrical version of a measuring instrument that disturbs nothing.
🔗 Linking question — in what ways can an electrical circuit be described as a system, in the same sense as the Earth's atmosphere or a heat engine? (Links to Topics B.2 and B.4: all three involve a flow — of charge, of energy, or of a working substance — through a defined system, driven by a difference: p.d., temperature difference, or pressure difference.)
Looking ahead: once current and p.d. are established, the next lesson asks what happens when a component resists that flow of charge — electrical resistance, \(R=V/I\), and Ohm's law.
⚠️ Trap corner – common mistakes
Misconception
Correct view
Current gets "used up" as it goes around a circuit
Charge is conserved: in a simple series loop, the current is the same at every point. Components use up energy, not charge
Conventional current is the direction electrons actually move
Conventional current runs from + to − by convention; the actual charge carriers in a metal, electrons, drift the opposite way, from − to +
A charge of, say, \(4.0\times10^{-19}\) C is possible
Charge is quantized: only whole-number multiples of \(e=1.60\times10^{-19}\) C occur (ignoring quarks)
An ammeter can be connected in parallel, like a voltmeter
Ammeters go in series (near-zero resistance, so as not to reduce the current); voltmeters go in parallel (near-infinite resistance, so as not to draw current)
"Voltage" is something that flows through a circuit
Current flows; potential difference doesn't flow — it's an energy difference between two points, measured across a component
⚠️ Most common slip: connecting an ammeter in parallel or a voltmeter in series "to make the wiring simpler." Because an ideal ammeter has ~0 Ω and an ideal voltmeter has ~∞ Ω, either mistake can short-circuit the supply or stop the current completely.
I = Δq/ΔtV = W/q1 V = 1 J C⁻¹1 C = 6.24×10¹⁸ e⁻charge is quantized: n × echarge is conserved (isolated system)ammeter: series, ≈0 Ωvoltmeter: parallel, ≈∞ Ωconventional current: + → −series p.d.s sum to supply V
Practice check — multiple choice
1. Which statement about electric charge is correct?
A Charge can take any value, however small
B Charge only occurs in whole-number multiples of \(e\)
C Only positive charge is quantized
D Neutrons carry a small positive charge
2. A steady current of 0.40 A flows for 25 s. The charge that has flowed is
A 0.016 C
B 6.25 C
C 10 C
D 25 C
3. An ideal voltmeter is connected across a resistor. What is true of its resistance and the current through it?
A Zero resistance, large current
B Zero resistance, no current
C Infinite resistance, large current
D Infinite resistance, (effectively) no current
4. Conventional current in a circuit flows
A from − to + inside the battery only
B in the same direction as electron drift
C from the + terminal, around the circuit, to the − terminal
D randomly, with no fixed direction
5. A 6.0 V battery drives a charge of 2.0 C around a circuit. The energy transferred by the battery is
A 3.0 J
B 12 J
C 0.33 J
D 8.0 J
6. Three components P, Q, R are in series with a 9.0 V battery. If \(V_{\text{P}}=2.0\) V and \(V_{\text{Q}}=3.5\) V, then \(V_{\text{R}}\) is
A 9.0 V
B 5.5 V
C 3.5 V
D 1.5 V
Practice check — fill in the blanks
Complete the statements:
Electric current is defined as the rate of flow of , \(I=\Delta q/\Delta t\), measured in . Charge can only exist in whole-number multiples of the elementary charge, so we say charge is . The total charge in an isolated system is always . Conventional current flows from the terminal of a battery, around the circuit, to the negative terminal. An ideal ammeter has resistance, and it is connected in . An ideal voltmeter has infinite resistance and is connected in . Potential difference is defined as work done per unit , \(V=W/q\).
Exam‑style questions
Exam question · Calculate / Explain
Q1. A carbon-12 atom contains six protons, six neutrons and six electrons. (a) Calculate the total positive charge in the atom. (b) State the total negative charge in the atom. (c) Explain what it means to describe the atom as neutral. (d) If the atom is ionized by the removal of one electron, state the charge on the ion. [4 marks]
✓ (a) \(6\times1.60\times10^{-19} = 9.6\times10^{-19}\) C. ✓ (b) \(-9.6\times10^{-19}\) C — the same magnitude, negative. ✓ (c) Neutral means zero net charge: the total positive and total negative charge are equal in magnitude, so they cancel. ✓ (d) Removing one electron leaves the atom with one unbalanced proton: charge \(=+1.60\times10^{-19}\) C.
Exam question · Calculate
Q2. \(2.5\times10^{20}\) electrons flow through a television in one minute. (a) Calculate the total charge which flows in that time. (b) Determine the electric current. [3 marks]
Q3. The charge flowing through a point on an overhead power-line pylon every hour is \(3.0\times10^{6}\) C. (a) Calculate the current. (b) Attractive pylon designs exist but cost more to build than standard designs. Suggest one reason people might be unwilling to pay for them. [3 marks]
✓ (a) \(I = \dfrac{3.0\times10^{6}}{3600}\) ✓ \(I \approx 8.3\times10^{2}\ \text{A}\) (830 A). ✓ (b) The extra cost would very likely be passed on to consumers through higher electricity prices, and many people prioritise lower bills over the visual appearance of infrastructure.
Exam question · Explain
Q4. Explain what is meant by a free or delocalized electron. [2 marks]
✓ An electron in a metal that has enough energy to no longer be bound to (attracted to) any one particular metal ion. ✓ Because these electrons are free to move through the lattice, they act as the mobile charge carriers that let metals conduct electric current well.
Exam question · Determine
Q5. At a junction, a current of 6 A enters at one branch and 5 A enters at a second branch. Three A leaves along one branch, and 1 A and 2 A leave along two further branches, with point P on the one remaining branch. Determine the current at P, stating its direction. [3 marks]
Fig 9 — Currents into and out of a junction. Conservation of charge means the total current in must equal the total current out.
✓ Conservation of charge at a junction: total current in = total current out. ✓ Current in \(= 6+5 = 11\) A; current out along the three known branches \(=3+1+2=6\) A. ✓ The remaining 5 A must leave through P, so \(I_{\text{P}} = 5\ \text{A}\), flowing out of the junction.
Exam question · Outline
Q6. Outline what would happen in a simple series circuit (cell, resistor, switch, ammeter) if the ammeter were replaced by a voltmeter. [2 marks]
✓ An ideal voltmeter has (near) infinite resistance, so almost no current can flow around the loop. ✓ The resistor would effectively switch off, and the voltmeter would read close to the full supply p.d. across itself.
Exam question · Explain
Q7. Explain why replacing the voltmeter connected across the battery terminals (measuring the supply p.d.) with an ammeter would be a bad idea. [2 marks]
✓ An ideal ammeter has (near) zero resistance, so connecting it directly across the battery terminals creates a short circuit. ✓ This would drive a very large current through the battery and ammeter, risking damage to both.
Exam question · State
Q8. In a series circuit with three components, the battery p.d. is 12 V. A voltmeter across the first component reads 2 V, and a voltmeter across the first two components together reads 5 V. State the p.d.s across the remaining components. [2 marks]
✓ P.d. across the second component \(= 5-2 = 3\ \text{V}\). ✓ P.d. across the third component \(= 12-5 = 7\ \text{V}\). (Note: only these two readings are mathematically determined from the three given voltmeter values — the source text's phrasing suggesting three unknown readings appears to be a slip.)
Exam question · Calculate
Q9. 400 C of electric charge flow through a lamp in one hour in a country where the mains supply is 230 V. (a) Calculate the current in the lamp. (b) Calculate the energy supplied from the mains to the lamp in this time. [3 marks]
