IB Physics B.5 Current and Circuits — Notes (SL & HL)

What provides the energy in a circuit?

Charge does not get “used up” in a circuit: the same charge carriers go round and round. What a circuit needs is a source of energy that does work on the charges, pushing them round the complete loop. In IB Physics that source is a cell (or a battery of cells), which is described by its electromotive force (emf).

• Emf is measured in volts: 1 V = 1 J C⁻¹. A 1.5 V cell gives 1.5 J to every coulomb that passes through it.
• Despite its name, emf is not a force; it is energy per unit charge.
• A battery is two or more cells connected together. Cells in series (+ to −) have a total emf equal to the sum of their emfs.

Chemical cells and solar cells

• In a chemical cell, chemical reactions between the electrodes and the electrolyte separate charge, transferring chemical energy into electrical energy. Primary cells are used once; secondary (rechargeable) cells, such as lithium-ion cells, can have their reactions reversed by pushing a current through them in the opposite direction.
• In a solar (photovoltaic) cell, light absorbed in a semiconductor junction releases charge carriers, transferring light energy directly into electrical energy. A single cell has a small emf (well under 1 V), so cells are connected in series and parallel to make panels. The current a panel can deliver increases with the intensity of the light falling on it.

How are circuits drawn, and where do meters go?

The diagram shows the arrangement of the components — what is connected to what — not their physical positions. Two diagrams that look different are the same circuit if they have the same connections. The symbols you are expected to recognise and draw are the ones listed in the data booklet:

• Draw wires as straight lines with right-angled corners. Mark a junction where wires join with a dot.
• An ammeter measures the current through a component, so it is connected in series with it.
• A voltmeter measures the pd across a component, so it is connected in parallel with it, one lead on each side.

• Unless a question says otherwise, meters are ideal. If a meter is non-ideal, its resistance is constant: treat it as an ordinary resistor in the circuit (see worked example 6).

What is electric current?

• \(I\) = current (A), \(\Delta q\) = charge passing the point (C), \(\Delta t\) = time taken (s). In the data booklet.
• The ampere is an SI base unit; the coulomb is defined from it: 1 C = 1 A s.
• Charge is quantised: every charge is a whole-number multiple of the elementary charge \(e=1.60\times10^{-19}\ \text{C}\) (data booklet). The number of electrons that make up a charge \(q\) is \(N=q/e\).
• Charge carriers are free particles that carry charge: electrons in metals, positive and negative ions in electrolytes and molten salts.
• Conventional current is drawn from the positive terminal to the negative terminal round the external circuit — the direction in which positive charges would flow. In a metal, the electrons flow the opposite way.

Conductors and insulators

Whether a material conducts depends on how many charge carriers it has that are free to move (mobile).

• The net drift of the electrons is slow (typically less than a millimetre per second), but a lamp lights almost instantly because the electric field is set up all round the circuit at once, and electrons everywhere in the circuit start drifting together.

What is potential difference?

• \(V\) = potential difference (V), \(W\) = work done, i.e. energy transferred (J), \(q\) = charge moved (C). In the data booklet.
• 1 volt = 1 joule per coulomb. If 2.0 C flows through a lamp with a pd of 9.0 V across it, \(9.0\times2.0=18\ \text{J}\) of electrical energy is transferred to internal energy and light.
• Positive charge moves from high to low potential through a resistor, transferring energy to it. Inside a source the charge is moved from low to high potential, gaining energy.

What is resistance and where does it come from?

• \(R\) in ohms (Ω); 1 Ω = 1 V A⁻¹. In the data booklet.
• \(R=V/I\) defines resistance at every point, for every component — ohmic or not. It is the ratio of the coordinates, never the gradient of a graph.

The origin of resistance

• The pd sets up an electric field in the conductor. Between collisions the field accelerates the free electrons, increasing their kinetic energy.
• The electrons repeatedly collide with the positive lattice ions (and with impurities and defects), transferring that energy to the lattice. The ions vibrate with larger amplitude: the internal energy and the temperature of the conductor increase. This is the heating effect of a current.
• The collisions limit the drift speed, which is why the current is finite for a given pd. More frequent collisions mean a smaller current for the same pd — a larger resistance.
• In a metal, a higher temperature means larger lattice vibrations and more frequent collisions, so the resistance of a metal increases with temperature.

Resistivity

• \(\rho\) = resistivity (Ω m), \(A\) = cross-sectional area (m²), \(L\) = length (m). In the data booklet; rearranged, \(R=\dfrac{\rho L}{A}\).
• For a wire of diameter \(d\), \(A=\pi\left(\dfrac{d}{2}\right)^2=\dfrac{\pi d^2}{4}\). Remember to halve the diameter and convert mm to m before squaring.

What is Ohm's law, and which conductors obey it?

• A conductor that obeys Ohm's law is ohmic: its resistance is constant, and its I–V graph is a straight line through the origin. A metal conductor at constant temperature is treated as ohmic.
• A conductor whose resistance changes with the current is non-ohmic. The standard example is a filament lamp.
• \(R=V/I\) is the definition of resistance and applies to everything. Ohm's law is the experimental observation that, for some conductors, \(R\) does not depend on \(V\) or \(I\).

Why the filament lamp is non-ohmic

• As the pd increases, the current increases and the power \(I^2R\) dissipated in the filament increases, so the filament's temperature rises (to well over 2000 °C in normal use).
• The lattice ions vibrate with larger amplitude, the electrons collide with them more frequently, and the resistance increases.
• So equal increases in \(V\) produce smaller and smaller increases in \(I\): the I–V curve bends towards the V axis.

The heating effect of resistors

Every resistor that carries a current warms up. A resistor designed for a circuit stays ohmic because it is made of a material such as constantan, whose resistivity hardly changes with temperature, and because it can lose energy to its surroundings fast enough to stay cool. If the current becomes large enough for its temperature to rise significantly, even a metal resistor stops being ohmic.

Two circuits are used to obtain I–V characteristics: a variable resistor in series with the component, or — better — a potentiometer used as a potential divider, which allows the pd to be varied all the way from zero.

How much power is dissipated in a resistor?

• \(P\) = power (W), \(I\) = current through the resistor (A), \(V\) = pd across it (V), \(R\) = its resistance (Ω). All three forms are in the data booklet.
• \(P=IV\) follows from \(V=W/q\) and \(I=q/t\): \(P=\dfrac{W}{t}=\dfrac{Vq}{t}=VI\). Substituting \(V=IR\) gives the other two forms.
• Energy transferred in time \(t\): \(W=Pt=VIt\).
• Choose the form that uses what is shared: in series the current is the same, so compare with \(I^2R\) (the larger resistor dissipates more power); in parallel the pd is the same, so compare with \(V^2/R\) (the smaller resistor dissipates more power).
• A lamp labelled “12 V, 24 W” dissipates 24 W when the pd across it is 12 V. Its operating resistance is \(R=V^2/P=6.0\ \Omega\). When cold, its resistance is much smaller, so the current at switch-on is larger than the operating current — which is why filament lamps often fail at the moment they are switched on.

How do you combine resistors in series and in parallel?

• All of these are in the data booklet.
• The combined resistance of a parallel combination is always smaller than the smallest resistance in it. Adding another branch in parallel always decreases the total resistance.
• \(n\) identical resistors \(R\): series \(nR\); parallel \(R/n\). For two resistors in parallel, \(R_p=\dfrac{R_1R_2}{R_1+R_2}\) (“product over sum”).
• In series the pds divide in the ratio of the resistances: the larger resistor takes the larger share of the pd.
• Method for a network: reduce each parallel group to a single resistance, add the series resistances, find the total current from the source, then work back outwards to find each pd and current.

What are emf and internal resistance?

A real cell is not a perfect source. The charge carriers must also pass through the materials inside the cell (the electrolyte and electrodes), which have resistance. This is the cell's internal resistance \(r\).

• \(\varepsilon\) = emf (V), \(I\) = current (A), \(R\) = external resistance (Ω), \(r\) = internal resistance (Ω). In the data booklet.
• Expanding: \(\varepsilon=IR+Ir\). The terminal pd (the pd across the external circuit) is \(V=IR\), so \[ V=\varepsilon-Ir \]
• \(Ir\) is the pd across the internal resistance — the energy per coulomb transferred to internal energy inside the cell, sometimes called the “lost volts”. A cell with a large current warms up.
• When no current flows (\(I=0\)), \(V=\varepsilon\): an ideal voltmeter connected directly across an isolated cell reads its emf.
• If the terminals are joined by a thick wire (\(R\approx0\)), the current is limited only by \(r\): the maximum (short-circuit) current is \(I=\varepsilon/r\).
• A car battery needs a very small internal resistance to deliver the hundreds of amperes needed by the starter motor; a high-voltage supply is given a large internal resistance on purpose, so that the current in an accidental short circuit is small (safety).

How do variable resistors and potential dividers work?

A resistor can have a variable resistance. In IB Physics the variable resistors are limited to three: the thermistor, the light-dependent resistor and the potentiometer. The data-booklet “variable resistor” symbol used in worked example 5 is simply a potentiometer track with only one end and the slider connected, so that it acts as an adjustable resistance.

Potential dividers

Two resistors in series across a supply share its pd in the ratio of their resistances. Taking the output across one of them gives a potential divider. Because the same current \(I=\dfrac{V_{in}}{R_1+R_2}\) flows through both (with nothing drawing current from the output):

• \(V_{in}\) = supply pd (V), \(V_{out}\) = output pd across \(R_2\) (V). This is not in the data booklet — derive it from the series rules each time, as here.
• If one of the resistors is a thermistor or an LDR, \(V_{out}\) changes with temperature or light level: the circuit becomes a sensor.
• A potentiometer connected across a supply is a continuously adjustable potential divider: the two parts of the track either side of the slider act as \(R_1\) and \(R_2\).
• If a meter or other load with a finite resistance is connected across the output, it is in parallel with \(R_2\), lowering the resistance of that part and so lowering \(V_{out}\). A voltmeter used here should have a resistance much larger than \(R_2\).

Exam tips for B.5

Common misconceptions

How this topic connects

Why resistors warm up: electrical work becoming internal energy

The pd does work on the charge carriers (\(W=qV\)), giving them kinetic energy between collisions. In collisions they pass this energy to the lattice ions, which vibrate with larger amplitude. In the kinetic model of B.1, this is an increase in the internal energy of the resistor, seen as a rise in temperature. The resistor then loses energy to its surroundings by conduction, convection and radiation (B.1), and it reaches a steady temperature when the rate of loss equals \(I^2R\). The same energy chain — work done → internal energy — is the one used for friction in A.3.

Electrons colliding with ions: a particle picture of resistance

Modelling a metal as a lattice of positive ions with a “gas” of free electrons explains resistance as collisions between the drifting electrons and the ions. The model correctly predicts that \(R\propto L\) (more collisions along a longer path), \(R\propto1/A\) (more parallel paths), that resistance rises with temperature (larger lattice vibrations), and that a current heats a conductor. Like every model it has limits: on its own it cannot explain why semiconductors behave the opposite way, which requires the extra idea that their number of free carriers rises with temperature — a good example of a model being extended when it meets new evidence.

Heat flow and charge flow: two conduction laws with the same shape

Thermal conduction obeys \(\dfrac{\Delta Q}{\Delta t}=kA\dfrac{\Delta T}{\Delta x}\) (B.1); electrical conduction obeys \(I=\dfrac{\Delta q}{\Delta t}=\dfrac{A}{\rho}\cdot\dfrac{V}{L}\). In both, a flow rate (of energy or of charge) is proportional to the area and to a “driving difference” per unit length (temperature gradient or potential gradient), with a material constant (\(k\) or \(1/\rho\)). Metals are good at both for the same reason: their free electrons carry charge and also transport energy quickly through the lattice. Insulators such as plastic are poor at both.

Key equations

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SL students: the previous SL page is B.3 Gas laws.