Electric Current, Voltage and Ohm's Law

The previous two chapters treated charges at rest. This one sets them moving. An electric current is charge in organised flow, and almost every practical use of electricity — heating, lighting, motors, electronics — is a matter of controlling how much charge flows, how fast, and how much energy each coulomb carries. Three quantities do that controlling: current $I$, potential difference $V$ and resistance $R$, linked for many materials by Ohm's law.

The chapter builds each quantity from a definition, connects the macroscopic picture (an ammeter reading, a voltmeter reading) to the microscopic one (electrons drifting through a lattice), and ends with the distinction between an ohmic and a non-ohmic conductor — a distinction the $I$–$V$ graph makes visible and that the next chapters on circuits depend on.

Reading note. The drift-velocity calculation in Section 3 gives a startling number — electrons crawl through a wire at well under a millimetre per second — and the resolution of that paradox (why a light comes on instantly) is worth understanding now, because it recurs whenever "the speed of electricity" is discussed.

1. Electric Current

Definition. The electric current through a cross-section of a conductor is the rate of flow of charge through it:

$$I = \frac{\Delta Q}{\Delta t}, \qquad \text{or in the limit} \qquad I = \frac{dQ}{dt}$$

The unit is the ampere ($\text{A}$), equal to one coulomb per second. Rearranged, the charge that passes in a time $t$ at steady current is $Q = It$.

Conventional current is defined as the direction in which positive charge flows — from the positive terminal of a supply, through the external circuit, to the negative terminal. In a metal the actual carriers are electrons, which drift the opposite way; the convention predates the discovery of the electron and is kept because it makes no difference to any circuit calculation.

Current is a scalar in circuit analysis, but it has a sense (a direction of flow) that must be tracked. At a junction, charge cannot accumulate, so the total current in equals the total current out — the basis of Kirchhoff's current law in the next chapter.

2. Potential Difference and EMF

Moving charge through a circuit requires energy, and that energy is delivered and spent in units of potential difference.

Definition. The potential difference (p.d.) between two points is the electrical energy converted to other forms per unit charge passing between them:

$$V = \frac{W}{Q}$$

The unit is the volt ($\text{V}$), equal to one joule per coulomb. A p.d. of $12\ \text{V}$ across a component means each coulomb passing through it delivers $12\ \text{J}$ of energy to that component.

The closely related electromotive force (emf, $\varepsilon$) of a source is the energy given to each coulomb by the source, converting chemical or mechanical energy into electrical energy. Emf and p.d. have the same unit and the same "energy per charge" form; the distinction — emf is energy in, p.d. is energy out — is developed fully in Chapter 6. For now, an ideal cell of emf $\varepsilon$ maintains a p.d. of $\varepsilon$ across its terminals.

3. The Microscopic Picture: Drift Velocity

Inside a metal, a "sea" of conduction electrons moves randomly at high speed ($\sim 10^{6}\ \dfrac{\text{m}}{\text{s}}$) with no net flow. Applying a p.d. superimposes a slow, steady drift on this random motion. If the wire has cross-sectional area $A$, carries $n$ free electrons per unit volume, each of charge $e$, drifting at speed $v_d$, then in a time $\Delta t$ every electron within a distance $v_d\Delta t$ of a cross-section passes through it. That is a volume $A v_d \Delta t$, containing $nA v_d \Delta t$ electrons and hence a charge $\Delta Q = neA v_d \Delta t$. Dividing by $\Delta t$:

Worked example. A copper wire of cross-sectional area $1.0\ \text{mm}^{2}$ carries a current of $3.0\ \text{A}$. Copper has $n = 8.5\times 10^{28}\ \text{m}^{-3}$ free electrons. Find the drift speed.

About $0.2\ \text{mm}$ per second. A light comes on instantly not because electrons race to it, but because the p.d. — and the electric field it sets up along the wire — is established through the whole circuit at close to the speed of light, so every electron, including those already in the filament, starts drifting almost at once.

4. Resistance

A given p.d. drives a larger current through some components than others. The property that quantifies this opposition is resistance.

Definition. The resistance of a component is the ratio of the p.d. across it to the current through it:

$$R = \frac{V}{I}$$

The unit is the ohm ($\Omega$), equal to one volt per ampere. Resistance arises microscopically from collisions between drifting electrons and the vibrating lattice ions; each collision transfers ordered kinetic energy to the lattice as heat, which is why a resistor warms up.

The resistance of a uniform wire depends on its geometry and material:

where $L$ is the length, $A$ the cross-sectional area and $\rho$ the resistivity of the material (unit $\Omega\cdot\text{m}$), a property of the substance and its temperature. Doubling the length doubles the resistance; doubling the area halves it.

5. Ohm's Law and $I$–$V$ Characteristics

Ohm's law. For a metallic conductor at constant temperature, the current through it is directly proportional to the p.d. across it:

$$V \propto I, \qquad \text{equivalently} \qquad R = \frac{V}{I} = \text{constant}$$

Ohm's law is not a universal law of nature — it is a statement that some components, over some range of conditions, have a constant resistance. Such components are ohmic. Their $I$–$V$ graph is a straight line through the origin, of gradient $\dfrac{1}{R}$.

Many important components are non-ohmic:

• A filament lamp: as current rises the filament heats, its resistance increases, and the $I$–$V$ graph curves so that the gradient decreases (the line bends towards the voltage axis).
• A semiconductor diode: conducts freely in one direction above a threshold p.d. (about $0.6\ \text{V}$ for silicon) and barely at all in reverse. The graph is strongly asymmetric.
• A thermistor (NTC type): resistance falls as temperature rises.

For any component, ohmic or not, $R = \dfrac{V}{I}$ still defines the resistance at a given operating point — it is just no longer a single fixed number.

6. Worked Example: A Two-Part Resistance Problem

A cylindrical carbon rod is $5.0\ \text{cm}$ long and $2.0\ \text{mm}$ in diameter, with resistivity $\rho = 3.5\times 10^{-5}\ \Omega\cdot\text{m}$. (a) Find its resistance. (b) A p.d. of $1.5\ \text{V}$ is applied across its ends; find the current and the drift speed, given $n = 2.0\times 10^{25}\ \text{m}^{-3}$.

(a) Cross-sectional area: $A = \pi r^{2} = \pi(1.0\times 10^{-3})^{2} = 3.14\times 10^{-6}\ \text{m}^{2}$.

(b) Current: $I = \dfrac{V}{R} = \dfrac{1.5}{0.557} = 2.69\ \text{A}$.

Drift speed:

The drift speed here is a thousand times larger than in the copper example because carbon has far fewer free electrons per unit volume — with fewer carriers, each must move faster to carry the same current.

Common Exam Pitfalls

• Confusing conventional current with electron flow. Conventional current runs from $+$ to $-$ in the external circuit; electrons drift the other way. State which you mean, and never mix them in one diagram.
• Quoting Ohm's law as $V = IR$ and calling every component ohmic. $R = V/I$ always defines resistance; Ohm's law is the extra claim that $R$ is constant, true only for a metal at fixed temperature.
• Reading the resistance off an $I$–$V$ curve as the gradient at a point. For a curved graph the resistance at an operating point is $\dfrac{V}{I}$ (the reciprocal of the chord gradient from the origin), not $\dfrac{dV}{dI}$.
• Forgetting the temperature condition. A filament lamp obeys $R = V/I$ at every instant but its $R$ changes as it heats, so it is non-ohmic.
• Using diameter for radius in $A = \pi r^{2}$, or leaving area in $\text{mm}^{2}$ — convert to $\text{m}^{2}$ ($1\ \text{mm}^{2} = 10^{-6}\ \text{m}^{2}$).
• Thinking current is "used up" in a component. Current is the same all the way round a single loop; it is energy per charge (p.d.) that is delivered and falls across each component.

Summary

• Current: $I = \dfrac{\Delta Q}{\Delta t}$, unit ampere; $Q = It$ for steady current. Conventional current is the direction of positive-charge flow, opposite to electron drift in a metal.
• Potential difference: $V = \dfrac{W}{Q}$, unit volt — energy converted per coulomb. Emf is the energy supplied per coulomb by a source; same unit, opposite bookkeeping.
• Microscopic current: $I = nAv_d e$. Drift speeds are tiny ($\sim 10^{-4}\ \dfrac{\text{m}}{\text{s}}$ in copper); the circuit responds fast because the field propagates near light speed.
• Resistance: $R = \dfrac{V}{I}$, unit ohm; for a uniform wire $R = \dfrac{\rho L}{A}$.
• Ohm's law: $V \propto I$ (constant $R$) for a metal at constant temperature — a straight-line $I$–$V$ graph through the origin. Filament lamps, diodes and thermistors are non-ohmic.