Current and circuits

An electric current is a flow of electric charge. In a metal wire the moving charges are free (conduction) electrons; in a solution or gas they can be ions. The current tells you how much charge passes a point each second:

$I = \frac{\Delta q}{\Delta t}$

where $I$ is the current in amperes (A), $\Delta q$ is the charge in coulombs (C) and $\Delta t$ is the time in seconds (s). One ampere is one coulomb per second ($1\,\text{A} = 1\,\text{C s}^{-1}$). Rearranged, the charge that flows is $\Delta q = I\,\Delta t$.

Charge is quantised. All charge comes in whole-number multiples of the elementary charge $e = 1.60\times10^{-19}\,\text{C}$ (the magnitude of the charge on one electron or proton). So the charge carried by $n$ electrons is $q = ne$, and one coulomb corresponds to $1/(1.60\times10^{-19}) \approx 6.25\times10^{18}$ electrons.

Conventional current vs electron flow. Long before the electron was discovered, scientists defined current as the direction in which positive charge would flow: from the + terminal to the − terminal in the external circuit. We now know that in metals it is the negatively charged electrons that actually move, and they drift the opposite way (from − to +). We keep the historical convention, so:

• Conventional current: + → − (used in all circuit analysis and formulae).
• Electron drift: − → + (the real motion of the charge carriers in a metal).

Measuring current and voltage. An ammeter is placed in series (in the current path) and ideally has near-zero resistance. A voltmeter is placed in parallel (across the component) and ideally has near-infinite resistance so it draws negligible current.

Charge flowing round a circuit carries energy. Two closely related "voltage" quantities describe this energy transfer per coulomb.

Potential difference (pd), or voltage, across a component is the electrical energy transferred to the component per unit charge passing through it: $V = \frac{W}{q}$ So $1\,\text{volt} = 1\,\text{joule per coulomb}$ ($1\,\text{V} = 1\,\text{J C}^{-1}$). When 1 C passes through a resistor with 3 V across it, 3 J of electrical energy is converted (into heat, in a resistor).

Electromotive force (emf), $\varepsilon$, of a source (a cell, battery or generator) is the energy supplied to each unit of charge by the source as it is driven round the circuit. Despite its name, emf is not a force — it is measured in volts, exactly like pd. The emf is the energy given out by the source per coulomb; the pd is the energy delivered to a component per coulomb.

Resistance measures how much a component opposes the current. It is defined as the ratio of the pd across the component to the current through it: $R = \frac{V}{I}$ Resistance is measured in ohms (Ω), where $1\,\Omega = 1\,\text{V A}^{-1}$. A large resistance means a small current for a given voltage.

Ohm's law. For a special class of components called ohmic conductors (e.g. a metal wire at constant temperature), the current is directly proportional to the pd: $V \propto I \quad\text{(temperature constant)}$ This means the resistance $R = V/I$ is constant, and a graph of $I$ against $V$ is a straight line through the origin. Ohm's law is not a universal law — it only holds while the temperature (and hence the resistance) stays constant. Components that obey it are ohmic; those that do not (like a filament lamp or a diode) are non-ohmic.

An I–V characteristic is a graph of current against potential difference for a component. Its shape tells you whether the component is ohmic and how its resistance changes.

1. Ohmic resistor (or a wire at constant temperature). The graph is a straight line through the origin. The current is proportional to the voltage, so $R = V/I$ is constant — the gradient of the line ($\Delta I/\Delta V$) equals $1/R$.

2. Filament lamp. As more current flows, the metal filament gets hotter. In a hot metal the lattice ions vibrate more, so the electrons collide with them more often — the resistance increases with temperature. The I–V graph therefore curves: it is steep near the origin (low resistance when cold) and flattens at higher voltages (higher resistance when hot). A filament lamp is non-ohmic.

3. Diode. A diode conducts current in one direction only. In the forward direction almost no current flows until the voltage exceeds a threshold (about 0.6–0.7 V for a silicon diode); above the threshold the current rises sharply. In the reverse direction essentially no current flows. A diode is strongly non-ohmic.

Reading resistance off the graph. For any component the resistance at a particular point is $R = V/I$ — the coordinates of that point, not the gradient (except for the special ohmic case, where they happen to agree). For the filament lamp, choose a point on the curve and divide its $V$ by its $I$; you will find $R$ is larger at higher voltages.

The combination rules are not things to memorise blindly — they follow directly from two conservation laws that hold in every circuit.

• Conservation of charge (a "junction" rule): charge is neither created nor destroyed, so the total current into any junction equals the total current out of it.
• Conservation of energy (a "loop" rule): each coulomb gains energy $\varepsilon$ from the source and must transfer all of it to the components, so the emf equals the sum of the potential differences round a loop.

Resistors in series (deriving $R_s = R_1 + R_2$). In series there are no junctions, so by conservation of charge the current $I$ is the same through both resistors. By conservation of energy the source pd is shared: $V = V_1 + V_2$. Using $V = IR$ for each part: $V = IR_1 + IR_2 = I(R_1 + R_2)$ Comparing with $V = IR_{s}$ for a single equivalent resistor: $\boxed{R_{s} = R_1 + R_2 + \dots}$ Resistors in series add — the total is always larger than any individual resistor.

Resistors in parallel (deriving $\tfrac{1}{R_p} = \tfrac{1}{R_1} + \tfrac{1}{R_2}$). In parallel both resistors connect the same two points, so they have the same pd $V$ across them. By conservation of charge the current splits at the junction: $I = I_1 + I_2$. Using $I = V/R$ for each branch: $I = \frac{V}{R_1} + \frac{V}{R_2} = V\left(\frac{1}{R_1} + \frac{1}{R_2}\right)$ Comparing with $I = V/R_{p}$: $\boxed{\frac{1}{R_{p}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots}$ Resistors in parallel give a total that is always smaller than the smallest individual resistor (adding a parallel path gives the current more routes, so the overall resistance drops).

Deriving the power equations. Electrical power is the rate of transferring energy, $P = W/t$. Since $V = W/q$, the energy transferred is $W = Vq$, and dividing by time with $I = q/t$: $P = \frac{W}{t} = \frac{Vq}{t} = V\cdot\frac{q}{t} = VI$ Now substitute Ohm's-law form $V = IR$: $P = VI = (IR)I = I^2R,\qquad\text{and}\qquad P = VI = V\!\left(\frac{V}{R}\right) = \frac{V^2}{R}$ $\boxed{P = IV = I^2R = \frac{V^2}{R}}$ Energy transferred in time $t$ is then $W = Pt = VIt$. Which form to use depends on what you know: use $I^2R$ when you know the current, and $V^2/R$ when you know the voltage.

A real cell is not a perfect source — the chemicals inside it also resist the current. This is the cell's internal resistance $r$. Because the current must flow through the cell as well as through the external circuit, some of the energy per coulomb is dissipated inside the cell before it ever reaches the external components.

The key equation. Applying conservation of energy round the whole loop, the emf (energy supplied per coulomb) equals the pd across the external resistor $R$ plus the pd across the internal resistance $r$: $\varepsilon = IR + Ir = I(R + r)$ The pd across the external circuit — the voltage you would actually measure at the cell's terminals — is called the terminal potential difference $V$: $V = IR = \varepsilon - Ir$ The term $Ir$ is often called the "lost volts": the energy per coulomb wasted inside the cell (usually as heat). So:

• When no current flows ($I = 0$, an open circuit), the lost volts are zero and the terminal pd equals the emf: $V = \varepsilon$. This is how you effectively measure emf — with a high-resistance voltmeter drawing negligible current.
• When a large current flows (small $R$), the lost volts $Ir$ are large, so the terminal pd drops well below the emf. This is why a torch dims when a heavy load is connected, and why a car's headlights dim as the starter motor turns.

A graphical trick. If you plot terminal pd $V$ against current $I$ (by varying the external resistance), rearranging $V = \varepsilon - Ir$ into the form $V = -rI + \varepsilon$ shows it is a straight line: the y-intercept is the emf $\varepsilon$ and the gradient is $-r$ (the negative of the internal resistance).

A potential divider is two (or more) resistors in series used to produce an output voltage smaller than the supply voltage. Because series resistors carry the same current and share the supply pd in proportion to their resistances, tapping off the voltage across one of them gives a controllable fraction of the input.

For a supply $V_{\text{in}}$ across resistors $R_1$ and $R_2$ in series, the current is $I = \dfrac{V_{\text{in}}}{R_1 + R_2}$. The output taken across $R_2$ is $V_{\text{out}} = IR_2$, so: $V_{\text{out}} = V_{\text{in}}\left(\frac{R_2}{R_1 + R_2}\right)$ The output is the supply voltage multiplied by the ratio of the resistance you tap across to the total resistance. Two useful checks:

• If $R_2$ is much larger than $R_1$, almost all the voltage appears across $R_2$, so $V_{\text{out}} \approx V_{\text{in}}$.
• If $R_2$ is much smaller than $R_1$, only a small fraction appears across $R_2$, so $V_{\text{out}}$ is small.

Why potential dividers are useful. If one of the resistors is a sensor whose resistance changes with conditions — a thermistor (resistance falls as temperature rises) or a light-dependent resistor / LDR (resistance falls as light increases) — then the output voltage changes with temperature or light. This makes potential dividers the basis of many sensing and control circuits.