Ohm's Law and Resistance Virtual Lab

Electric current is a flow of charge, and every practical circuit, from a torch to a national grid, depends on three linked quantities: current I, potential difference V (p.d.) and resistance R. This laboratory lets you build circuits on a bench, read ammeters and voltmeters, record data and discover the relationships for yourself. The central relationship is Ohm's law: for an ohmic conductor at constant temperature, the current is proportional to the potential difference across it. Almost everything else in the topic (the series and parallel rules, power, resistivity, internal resistance) follows from that idea together with conservation of charge and conservation of energy.

The notes work in three layers. Material marked Foundation needs basic algebra and the idea of a circuit. Core material matches typical A-level, IB Diploma, AP Physics 1, WASSCE and Cambridge AS and A Level courses. Extension material goes towards IB HL, AP Physics C: Electricity and Magnetism and first-year university work, and uses calculus ideas and a little statistics.

The default bench setting is a 100 Ω resistor on a 6 V supply, which draws 60 mA. Before you change anything, predict what the meters will show, then test your prediction. When these notes quote a number from the simulator for a lamp or a diode, that number belongs to the simulator's model, which is described in section J. Real components differ in detail.

Learning outcomes

Foundation

• Define current as the rate of flow of charge and potential difference as energy transferred per unit charge, and state their units.
• Connect an ammeter in series and a voltmeter in parallel, and read them correctly.
• Use V = IR to find any one of the three quantities from the other two.
• State that in a series circuit the current is the same everywhere and in a parallel circuit the p.d. is the same across each branch.
• Describe how a filament lamp and a diode differ from an ohmic resistor.
• Explain why a fuse protects a circuit and why mains electricity is dangerous.

Core (A-level, IB, AP, WAEC)

• Add resistances in series and combine them in parallel, and find the p.d. and current for every component.
• Use P = IV = I²R = V²/R and energy in kilowatt-hours.
• Use R = ρL/A and explain how resistivity differs from resistance.
• Interpret I-V characteristics for an ohmic resistor, a filament lamp and a semiconductor diode, and explain the shape of each.
• Use emf and internal resistance, V = ε − Ir, and find both from a graph.
• Plan an investigation, record data in a table, plot a graph with a line of best fit and state the uncertainty in a gradient.

Extension (HL, AP C, university)

• Derive I = nAvq and estimate drift speed.
• Derive the condition for maximum power transfer, R = r, and the efficiency at that point.
• Analyse a loaded potential divider and explain loading by meters.
• Use standard errors and 95% intervals for gradients and intercepts, and compare chord with tangent (differential) resistance for a non-ohmic device.

Current Foundation

Current is the rate of flow of charge: I = ΔQ/Δt. The unit is the ampere (A), one coulomb per second. By convention the current is drawn from the positive terminal of a source round the circuit to the negative terminal. This is conventional current. In a metal the charge carriers are electrons, which drift the opposite way. The simulator's moving dots show conventional current.

Potential difference and emf Foundation

The potential difference between two points is the energy transferred per unit charge as charge moves between them: V = W/Q. The unit is the volt (V), one joule per coulomb. A cell's emf ε is the energy the cell supplies per unit charge passing through it, including the energy wasted inside the cell. The p.d. across the terminals is smaller than the emf whenever current flows, because some energy is lost against the cell's internal resistance. Section C and the simulator's cell mode show this.

Resistance Foundation

Resistance is the ratio of p.d. to current: R = V/I. The unit is the ohm (Ω), one volt per ampere. This is a definition and it works for any component at any point on its characteristic. A large resistance means a small current for a given p.d.

Ohm's law Foundation

Ohm's law is a statement about behaviour, not a definition: the current through an ohmic conductor is directly proportional to the p.d. across it, provided the temperature and other physical conditions stay constant. For such a conductor V/I is the same at every point, so the graph of V against I is a straight line through the origin whose gradient is R. Metals at constant temperature are ohmic. Filament lamps, diodes, thermistors and light-dependent resistors are not. They still have a resistance V/I at each point, but it changes.

Kirchhoff's laws Core

The current law says that the total current into a junction equals the total current out, because charge is conserved. The voltage law says that around any closed loop the sum of the emfs equals the sum of the p.d.s, because energy is conserved. These two laws, plus V = IR for each component, solve every circuit in this lab.

Power and energy Core

Power is the rate of energy transfer. When charge Q falls through a p.d. V it transfers energy QV, so P = QV/t = IV. Combining with V = IR gives P = I²R = V²/R. Electrical energy is also sold in kilowatt-hours: 1 kWh = 3.6 MJ. The heating in a resistor is the energy that the charge carriers hand to the lattice in collisions.

Resistivity Core

The resistance of a uniform wire depends on its material and its shape. Doubling the length doubles the resistance, and doubling the cross-sectional area halves it, so R = ρL/A. The constant ρ is the resistivity of the material, in ohm metres (Ω m). Resistance describes an object. Resistivity describes a material. Copper has a resistivity of about 1.7 × 10^-8 Ω m, constantan about 4.9 × 10^-7 Ω m and nichrome about 1.1 × 10^-6 Ω m, which is why nichrome and constantan are used for heating elements and precision resistors while copper is used for cables.

Series resistors Core

The same current I flows through each resistor. By the voltage law the supply p.d. is the sum of the p.d.s: V = V₁ + V₂ + V₃. Substituting V = IR for each term gives IR = IR₁ + IR₂ + IR₃. Dividing by I gives R = R₁ + R₂ + R₃.

Parallel resistors Core

Every branch has the same p.d. V. By the current law the total current is I = I₁ + I₂ + I₃. Substituting I = V/R for each term gives V/R = V/R₁ + V/R₂ + V/R₃. Dividing by V gives 1/R = 1/R₁ + 1/R₂ + 1/R₃. For two resistors this is R = R₁R₂/(R₁ + R₂), and for n identical resistors it is R/n.

Resistivity from proportionality Core

Imagine a wire cut into n identical pieces joined in series: its resistance is n times that of one piece, so R ∝ L. Imagine n identical wires laid side by side and joined at both ends (in parallel): the resistance is one nth, so R ∝ 1/A. Combining gives R = ρL/A with ρ a constant of the material at a given temperature. For a round wire A = πd²/4, so doubling the diameter divides the resistance by four.

Current and drift speed Extension

Let a conductor of cross-sectional area A contain n charge carriers per unit volume, each of charge q, drifting at mean speed v. In time Δt the carriers in a length vΔt pass a cross-section. That volume is AvΔt, it holds nAvΔt carriers and so a charge nAvqΔt. Dividing by Δt gives I = nAvq. For copper n is about 8.5 × 10²⁸ per cubic metre, so a current of 2 A in a wire of diameter 1 mm gives v = I/(nAq) = 2 / (8.5 × 10²⁸ × 7.85 × 10^-7 × 1.60 × 10^-19) ≈ 1.9 × 10^-4 m/s. The electrons creep along at under a fifth of a millimetre per second. The lamp lights at once because the electric field, which sets all the carriers moving, travels along the wire at a large fraction of the speed of light.

Cell with internal resistance Core

A real cell behaves like an ideal source of emf ε in series with a small resistance r. With an external resistance R the total resistance is R + r, so ε = I(R + r). The p.d. across the terminals is the p.d. across R: V = IR = ε − Ir. The term Ir is sometimes called the lost volts. Compared with y = mx + c, a graph of V (vertical) against I (horizontal) has intercept ε and gradient −r.

Maximum power transfer Extension

The power in the load is P = I²R = ε²R/(R + r)². Differentiating, dP/dR = ε²[(R + r) − 2R]/(R + r)³, which is zero when R = r. The maximum power is ε²/4r, and at that point the efficiency R/(R + r) is exactly 50%. For example a 9 V cell with r = 3 Ω delivers at most 81/12 = 6.75 W, to a 3 Ω load. The simulator's power graph shows this peak.

Potential divider Core

Two resistors R₁ and R₂ in series across V_in carry the same current I = V_in/(R₁ + R₂). The p.d. across R₂ is IR₂, so V_out = V_inR₂/(R₁ + R₂). This holds only if nothing draws current from the output. A load in parallel with R₂ lowers the output (worked example 7).

Every number below was checked by computation. Try each one yourself first, then compare. Several can be reproduced on the bench or in the prediction calculator in the Analyse step.

Choosing the axes Core

Examination papers and textbooks draw characteristics in two ways. An I-V graph has current on the vertical axis and p.d. on the horizontal axis. The gradient is then 1/R (the conductance) for an ohmic component. A V-I graph, the one used in this lab's tool, has p.d. on the vertical axis and the gradient is R. Always read the axis labels before you interpret a gradient. A rising line on an I-V graph that bends upward means the resistance is falling. A line that bends upward on a V-I graph means the resistance is rising.

The three characteristics Core

• Ohmic resistor: a straight line through the origin. The gradient of the V-I graph is the constant resistance.
• Filament lamp: the line curves so that V/I rises with current. A larger current heats the filament, the lattice ions vibrate more and collide more often with the electrons, and the resistance rises. In the lab's lamp model the resistance is 4.2 Ω at 1 V and 12 Ω at 6 V.
• Semiconductor diode: almost no current until the forward p.d. reaches about 0.6 to 0.7 V for silicon, then the current rises very steeply, with a protective resistor needed to limit it. In reverse bias only a tiny leakage current flows. The V-I graph through the origin shows a long near-vertical rise and then an almost flat stretch: on a V-I graph the diode's p.d. barely changes while the current grows. In the lab, a 3 V supply drives 22.6 mA through a diode with its 100 Ω protective resistor and the p.d. across the diode alone is 0.74 V.

Chord and tangent resistance Extension

For a non-ohmic component the word 'resistance' needs care. The chord or static resistance at a point is V/I at that point, the slope of the line from the origin. The tangent or differential resistance is dV/dI, the slope of the curve at that point. For an ohmic resistor they are equal. For a lamp the tangent value exceeds the chord value. For a diode in forward conduction the tangent resistance is far smaller than the chord value. The analysis tool reports both for a lamp or diode.

Cell graph Core

Plot terminal p.d. V against current I for a cell supplying different loads. Because V = ε − rI the points lie on a straight line with a negative gradient. The intercept on the V axis is the emf (the p.d. when no current flows). The intercept on the I axis would be the short-circuit current ε/r, but you should never actually short-circuit a real cell to find it (section J).

Resistance against length Core

For a wire of fixed material and diameter, R plotted against L is a straight line through the origin with gradient ρ/A. Measure the diameter with a micrometer, work out A, and then ρ = gradient × A. A positive intercept on the R axis suggests contact or lead resistance that does not scale with length.

Power against load Extension

Plotting the load power against R for a cell gives a curve that rises, peaks at R = r and falls slowly. The curve is asymmetric: very large loads draw little current, while very small loads waste most of the power inside the cell.

Temperature Core

Ohm's law holds only when the temperature is steady. In a metal a rise in temperature increases the vibration of the lattice ions, so the conduction electrons collide more often and the resistivity increases. A tungsten lamp filament is hundreds of degrees hot when lit, and its resistance is many times its cold value. This is why a lamp draws a large surge of current in the first moments after it is switched on. In a semiconductor such as the material of a thermistor, heating releases more charge carriers, which usually outweighs the extra collisions, so the resistance of an NTC thermistor falls as it warms. In a few materials at very low temperatures the resistance drops to zero. These are superconductors.

Heating of the test component Core

Even a resistor warms when current flows, because P = I²R. A fine wire can warm enough during an experiment to change its resistance. For that reason, when you test a wire for ohmic behaviour you use small currents, take readings quickly and switch off between them. The simulator keeps resistor and wire resistance constant, which is why they appear ideally ohmic. In a real laboratory a plot for a heavily loaded wire curves upward.

Meters are not ideal Core

An ideal ammeter has zero resistance and an ideal voltmeter has infinite resistance. Real instruments fall short. A moving-coil ammeter or a milliamp range can add an ohm or more in series, and a voltmeter draws a small current. In low-resistance circuits the ammeter matters. In high-resistance circuits the voltmeter matters, because it forms a parallel path and lowers the apparent resistance. Digital multimeters usually have a voltmeter resistance of several megohms. The lab's meters are treated as ideal.

Leads, contacts and tolerance Core

Connecting wires and crocodile clips add small resistances, which matter when the circuit resistance is only a few ohms (for example when measuring the internal resistance of a cell). Resistors are sold with a tolerance: a 470 Ω resistor marked ±5% may be anywhere from 446.5 Ω to 493.5 Ω. A measured gradient that differs from the printed value by less than the tolerance is not a discrepancy.

Real sources Core

The model of an emf in series with a constant r is only approximate. The internal resistance of a real cell rises as it discharges and falls if the cell is warm. A cold lead-acid battery in a car delivers a smaller current than a warm one. Cells of the same type from different batches may have different internal resistances. A supply labelled 5 V from a bench power supply may be regulated so that its terminal p.d. holds steady until a current limit, then collapses.

Direct current only Extension

Everything in this lab is direct current (DC). Alternating current brings in capacitance and inductance as well, and a quantity called impedance replaces resistance. The mains 230 V figure used in some examples is the root-mean-square value, which gives the same heating effect in a resistor as a steady DC p.d. of that size.

Variables Foundation

Every investigation has an independent variable that you change, a dependent variable that you measure, and control variables you hold constant. In an Ohm's law experiment the independent variable is the p.d. (set by the supply), the dependent variable is the current, and the controls include the temperature of the component, its length and its material. If the temperature is not controlled, the result is a hidden second variable.

Investigation 1: is a component ohmic? Core

1. Connect the component, an ammeter in series and a voltmeter directly across the component. (On the bench, set Single component.)
2. Start with the supply at its smallest non-zero value and increase it in equal steps. A sweep of ten points from one tenth of the maximum to the maximum is a good plan. Use at least six values.
3. For each setting record I and V. Switch off or reduce the supply between readings if the component warms.
4. Reverse the supply and repeat if the component might behave differently in the two directions (a diode does).
5. Plot V against I. A straight line through the origin shows ohmic behaviour. Use the gradient for R.

Investigation 2: resistivity of a wire Core

Measure the diameter of the wire at several places with a micrometer, taking readings in two perpendicular directions, and take the mean. Fix one end of the wire and move a contact along it to vary the length L. For each length record V and I at a small, constant current and find R = V/I. Plot R against L. The gradient is ρ/A, so ρ = gradient × A with A = πd²/4. In the lab, choose Resistance wire, keep the material and diameter fixed, and record at several lengths.

Investigation 3: emf and internal resistance Core

Connect a cell to a variable resistor (rheostat) and an ammeter in series, with a voltmeter across the cell. Start with the largest resistance so the current is small, and increase the current in steps. Record V and I, and keep each reading brief so the cell does not discharge. Plot V against I: the intercept is ε and minus the gradient is r. Use a fresh cell if a long experiment is planned, and never connect a cell directly across its own terminals.

Investigation 4: series and parallel rules Foundation

Choose three resistors of known values. Measure the total current and supply p.d. for the series arrangement and for the parallel arrangement. Compare V/I with the sum for series and with the reciprocal rule for parallel. Resistor tolerance (5% is common) limits how closely they should agree.

Range, repeats and risk Core

Choose a range that gives a good spread of points while keeping currents below the component's rating. Repeat the whole set if you can, to check for scatter. List the risks (hot wires, battery short circuit, mains) and say how you will control them. A good plan names the apparatus with their ranges and resolutions, explains how each reading will be taken and says how the data will be analysed.

Recording data Foundation

Record readings in a table with each column headed by the quantity and its unit, such as I / mA. Give every reading in a column to the same number of decimal places, matching the resolution of the instrument. The simulator's CSV download follows this pattern. Quote final answers to a sensible number of significant figures, usually two or three, no more than the data justify.

Absolute and percentage uncertainty Core

Every measurement has an uncertainty. For a single reading it is often taken as half the smallest division or the last digit on a digital display. The absolute uncertainty has the units of the quantity, for example 0.05 V. The percentage uncertainty is the absolute uncertainty divided by the value, times 100. When quantities are multiplied or divided, add the percentage uncertainties. For R = V/I, the percentage uncertainty in R is the percentage in V plus the percentage in I.

For a quantity raised to a power, multiply the percentage uncertainty by the power. A diameter measured to 2% gives an area, and therefore a resistance, uncertain by about 4%, since R ∝ 1/d². This is why the diameter is the most important measurement in a resistivity experiment.

Lines of best fit Core

A graph averages out random error. Draw the line of best fit so that points are scattered evenly above and below it. Then find the gradient by choosing two points on the line (not data points) far apart, and quoting the gradient from Δy/Δx with units. To estimate its uncertainty by hand, draw the steepest and shallowest lines that still pass through all the error bars or the scatter of the points. The uncertainty in the gradient is half the difference between their gradients. The intercept is treated the same way.

Least squares and standard errors Extension

Software fits a line y = a + bx by minimising the sum of the squared vertical distances (the residuals). For n points the standard error in the gradient is s/√S_xx, where s² is the sum of squared residuals divided by n − 2, and S_xx = Σ(x − mean x)². A 95% interval is the standard error times the Student t value for n − 2 degrees of freedom. With only five points t is 3.18, which is why a small data set has a wide interval. If the fit is forced through the origin there are n − 1 degrees of freedom. The tool in the Analyse step reports the gradient with its standard error and its 95% half-width, and draws two dashed lines for the gradient plus and minus one standard error.

As a check, take the data in worked example 8 of section D. The gradient is −0.480 with a standard error of 0.016, so 0.480 ± 0.016 is about 3.4% in r. Five points scattered by about 0.01 V give a gradient good to a few per cent. More readings over a wider current range would improve this.

Is the intercept consistent with zero? Extension

For a resistor, V = IR predicts zero intercept. A fitted intercept of (0.015 ± 0.038) V is consistent with zero because it is smaller than its own standard error. An intercept clearly different from zero signals a systematic problem such as a zero error on a meter, extra series resistance, or an offset in the supply. The same applies to the R-against-L graph.

Comparing with an accepted value Core

Calculate the percentage difference between your result and the accepted or set value. If the difference is smaller than your estimated uncertainty, the results agree within uncertainty. If it is larger, look for a systematic error rather than blaming random scatter. Averaging many readings reduces random error but cannot remove a systematic one.

Connecting meters Foundation

An ammeter goes in series, so that the same current flows through it as through the component. A voltmeter goes in parallel, directly across the component whose p.d. you want. Reversing a digital meter gives a negative reading. An analogue meter may be damaged. When you use a multimeter, set the range above the largest reading you expect and move down, so that a reading is never off scale.

Resolution and range Core

The resolution is the smallest change the meter can show. A digital meter reading 0.250 A has a resolution of 0.001 A. A scale with 0.1 V divisions has a reading uncertainty of about 0.05 V. Choosing a range that makes the reading fill most of the scale reduces the percentage uncertainty. A 0.05 V uncertainty on a 5 V reading is 1%, but on a 0.5 V reading it is 10%.

Variable supply: rheostat or divider Core

To vary the current through a component there are two ways. A rheostat in series changes the total resistance, so the current can only fall from the maximum, and the minimum is not zero. A potential divider uses a variable resistor across the supply with the component connected across part of it, so the p.d. across the component can be set from zero up to the full supply. The divider is preferred for characteristic curves of lamps and diodes. The simulator offers an ideal variable supply that behaves like a divider.

Resistor colour code Foundation

Carbon and metal film resistors use coloured bands: black 0, brown 1, red 2, orange 3, yellow 4, green 5, blue 6, violet 7, grey 8, white 9. The first two bands (or three on a precision resistor) give the digits, the next is the multiplier (a power of ten), and the last gives the tolerance (gold ±5%, silver ±10%). Brown, black, red, gold means 10 × 10² = 1000 Ω, or 1.0 kΩ, at ±5%. Check the manufacturer's key because variants exist.

Measuring wire diameter Core

Use a micrometer screw gauge, which reads to 0.01 mm. Check for a zero error by closing the jaws and subtracting the zero reading. Measure in several places along the wire and at right angles to each other, since the wire may not be perfectly round or uniform. The mean diameter gives A = πd²/4.

Four-terminal measurement Extension

Low resistances (a fraction of an ohm) are hard to measure with two leads because the lead and contact resistances add to the result. In a four-terminal (Kelvin) method one pair of leads carries the current and a separate pair senses the p.d. across the sample. The sensing leads carry almost no current, so their resistance has almost no effect. Data loggers and modern multimeters use this idea to measure very small resistances.

What this simulation models Foundation

• An ideal variable DC supply from 0 to 12 V with ideal meters (an ammeter with zero resistance and a voltmeter with infinite resistance).
• Ohmic resistors and wires with fixed resistance at constant temperature. Wire resistance is R = ρL/A with the resistivities shown on the controls.
• A filament lamp described by the empirical model V = R₀I(1 + (I/I₀)²) with R₀ = 2 Ω, chosen so that it draws 0.5 A at 6 V (12 Ω when lit). This reproduces the rise of resistance with current but is not a heat-transfer model.
• A silicon diode following the Shockley equation with saturation current 10^-12 A and ideality factor 1.2, in series with a 100 Ω protective resistor, and no breakdown in reverse.
• Series and parallel networks and a cell with a constant internal resistance, solved exactly with Ohm's law and Kirchhoff's laws. An optional random scatter of about 0.5% mimics reading error.

Real-world safety Foundation

The simulator is harmless. Real circuits are not, and the habits you practise here should carry over safely.

• Never experiment with mains electricity. A mains supply (230 V in many countries, 120 V in others) can deliver a current through the body that stops the heart. Only use low-voltage supplies (typically 12 V or less) in school and college experiments, from isolated laboratory sources.
• A cell or battery has a low internal resistance and can supply a very large current if its terminals are joined by a wire. A short circuit can overheat the wire, burn skin, and cause a battery to vent or ignite. Lithium batteries and lead-acid car batteries are especially hazardous. Do not short-circuit them and do not charge disposable cells.
• A resistor, lamp or wire carrying a large current becomes hot enough to burn. Check ratings. Let components cool before handling.
• A fuse is a deliberately weak link that melts when the current exceeds its rating, breaking the circuit. In a mains plug it is in the live wire. A residual-current device (RCD) cuts the supply within milliseconds if the current in the live and neutral wires differs, which can happen if current flows through a person to earth. An RCD that trips at about 30 mA is typical in many countries.
• Do not use damaged leads, do not use wet hands near electrical equipment, and switch off before changing a circuit. Follow your teacher's instructions and the local safety rules of your school or laboratory.

The notes give general information for learning and do not replace training, a risk assessment or the safety rules where you work.

1. Confusing Ohm's law with the definition of resistance. R = V/I always defines resistance. Ohm's law says that for some conductors it is constant. A lamp has a resistance at every point, but it does not obey Ohm's law.
2. Wrong meter connection. An ammeter in parallel is almost a short circuit. A voltmeter in series nearly stops the current.
3. Adding resistors in parallel. Parallel resistors do not add. The combined resistance is smaller than the smallest branch. If your answer is bigger, you have forgotten to take the reciprocal at the end.
4. Using the gradient of an I-V graph as R. With current on the vertical axis the gradient is 1/R.
5. Mixing up emf and terminal p.d. Terminal p.d. is lower than emf whenever current flows. They are equal only for an open circuit or zero internal resistance.
6. Forgetting the internal resistance in series. The current from a cell is ε/(R + r), not ε/R.
7. Unit errors in resistivity. Using millimetres or the diameter directly instead of the area in square metres. Convert 0.40 mm to 0.40 × 10^-3 m and square it.
8. Treating resistance and resistivity as the same. Resistance depends on shape. Resistivity does not.
9. Believing current is used up. The current is the same all round a series loop. Energy is transferred, not charge lost.
10. Power formula slips. Use P = V²/R only with the p.d. across that resistor, and P = I²R with the current through it.
11. Drawing the line through the origin by habit. Only force the line through the origin if the theory demands it. Otherwise report the intercept.
12. Rounding too early. Keep extra digits in intermediate steps and round the final answer only.

Command words Foundation

Read the verb carefully. State wants a short fact. Define wants a precise statement (for example, potential difference is the energy transferred per unit charge). Calculate wants working and a numerical answer with a unit. Explain wants a reason, usually a chain of linked ideas. Show that gives you the answer, and the marks are for the working, so every step must appear. Suggest expects you to apply knowledge to an unfamiliar situation.

Working and units Core

Write the formula first, substitute with units, then give the result. Examiners can award method marks even when the final answer is wrong. Give the unit every time, with the correct prefix (mA, kΩ). A final answer is typically expected to the number of significant figures of the least precise data, often two or three.

Typical question patterns Core

• Circuit with several resistors: reduce parallel groups to a single resistance first, add the series parts, then find the total current and work back to find the p.d. and currents in each part.
• Lamp characteristic: describe the shape, state that resistance rises, and explain in terms of lattice vibrations and electron collisions.
• Internal resistance: start from ε = V + Ir and match to y = mx + c.
• Resistivity practical: say what you would plot, what the gradient means, and which measurement dominates the uncertainty (the diameter).
• Power in a heater: choose the formula that uses the quantities given.

Graph questions Core

Label axes with quantity and unit, choose scales that use more than half of the grid, plot points to the nearest half square and draw one clear best-fit line or smooth curve. Use a large triangle for the gradient and show its coordinates. Say what the gradient and the intercept mean in terms of the physics.

Explanations that gain marks Extension

An explanation of a filament lamp's characteristic that earns full marks usually has three links: the current heats the filament, the lattice ions vibrate with greater amplitude, and the electrons collide more often so the resistance rises. An explanation of the diode should mention the forward threshold, the rapid rise in current once it is exceeded, and the very small reverse current.

Using the lab to prepare Foundation

Work through the guided tasks, then try the practical exam. The review list that follows tells you which note section to revisit. Marks in this lab are for learning, not an official grade.

Courses differ in order, depth and notation. The lists below give topic names only, so that you can see where this laboratory fits. They are not a statement of any specification. Always check your own current specification, syllabus or course outline for the exact content, the equations you are given and the practical skills assessed.

Use the three level tags to match material to your course. Foundation items suit introductory and lower secondary study. Core items match most upper secondary and first-year university courses. Extension items reach towards higher-level and university work. This laboratory supports your course and does not replace it.

Heating elements Core

Kettles, toasters and electric ovens use a wire of high resistivity, often a nickel-chromium alloy, because it heats strongly and does not oxidise quickly at high temperature. For a 2.3 kW kettle at 230 V the element has a resistance of 23 Ω and carries 10 A. The same current in the supply cable should produce little heating, so cables use thick copper with a tiny resistance.

Cables and energy loss Core

A 20 m run of twin copper cable (40 m of conductor) with a cross-section of 2.5 mm² has a resistance of 1.68 × 10^-8 × 40 / 2.5 × 10^-6 = 0.27 Ω. At 13 A the cable dissipates I²R = 45 W as heat and drops 3.5 V. National grids send power at very high voltage so that the current, and the I²R loss, is small for a given power.

Fuses and breakers Foundation

A fuse wire melts when its current rating is exceeded, because the heating I²R becomes too large. A circuit breaker uses a heating strip or an electromagnet to open a switch. Both protect wiring from overheating.

Sensors Core

A thermistor in a potential divider turns temperature into a p.d., and a light-dependent resistor does the same for light. A strain gauge is a thin wire whose resistance changes slightly when it is stretched, since both its length and cross-section change. These sensors all use V = IR and the divider rule.

Batteries and electric vehicles Extension

Internal resistance limits the current a battery can deliver and wastes energy as heat inside it. Engineers choose cells and pack layouts with low internal resistance for high-power uses. Cells in series add emf and internal resistance. Identical cells in parallel keep the emf but divide the internal resistance.

Diodes Core

Diodes allow current in one direction only. They rectify alternating current, protect circuits from reversed supplies and, as light-emitting diodes, give efficient lighting, always with a series resistor to set the current.

Microscopic form of Ohm's law Extension

In the free-electron model, the average drift speed is v = (eE/m)τ, where E is the electric field and τ is the mean time between collisions. Using I = nAve gives a current density J = (ne²τ/m)E = σE, with conductivity σ = 1/ρ. This is the microscopic Ohm's law. Heating raises the collision rate, shortening τ, which explains why the resistivity of a metal rises with temperature.

Temperature coefficient Extension

Over modest ranges the resistance of a metal follows R = R₀(1 + αΔT). For copper α is about 0.0039 per kelvin, so heating from 20 °C to 100 °C raises the resistance by a factor of 1 + 0.0039 × 80 = 1.31.

Diode equation Extension

The ideal diode current is I = I_s(e^V/nV_T − 1), with thermal voltage V_T about 25.9 mV at room temperature. The current grows exponentially with p.d., so a change of about 60 mV multiplies it by roughly ten when n = 1. That is why the forward p.d. of a silicon diode stays near 0.6 to 0.7 V across a huge range of currents.

Loaded dividers and Thévenin's idea Extension

Any network of resistors and sources, seen from two terminals, behaves like a single emf in series with a single resistance. For the divider in worked example 7 the equivalent source is 6.0 V with 500 Ω, so a 1.0 kΩ load gets 6.0 × 1000/1500 = 4.0 V. This is the same mathematics as a cell with internal resistance.

Statistics of the fit Extension

Weighted fits give less influence to noisier points. Residual plots show whether a straight line is the right model: a systematic curve in the residuals means it is not. Report the number of points, the fit method and the uncertainty.

• I = ΔQ/Δt, V = W/Q, R = V/I. Resistance is defined for every component at every point.
• Ohm's law: current is proportional to p.d. for an ohmic conductor at constant temperature. The V-I graph is a straight line through the origin with gradient R.
• Ammeter in series, voltmeter in parallel.
• Series: same current, p.d.s add, resistances add. Parallel: same p.d., currents add, 1/R adds. Parallel total is smaller than the smallest branch.
• P = IV = I²R = V²/R. 1 kWh = 3.6 MJ.
• R = ρL/A. Resistance depends on the object, resistivity on the material.
• A filament lamp is non-ohmic because its resistance rises as it heats. A diode conducts one way above a threshold of about 0.6 to 0.7 V.
• Cell: ε = I(R + r) and V = ε − Ir. Plot V against I: intercept ε, gradient −r.
• Maximum power transfer occurs when R = r, at 50% efficiency.
• Quote a gradient with its uncertainty, and compare it with the accepted value within that uncertainty.
• Never experiment with mains electricity. Never short-circuit a battery.

Ampere (A)

The unit of current, one coulomb per second.

Charge carrier

A particle that carries current, such as an electron in a metal.

Chord resistance

V/I at a point on a characteristic.

Conductance

The reciprocal of resistance, in siemens (S).

Conventional current

Current drawn from the positive terminal round the circuit to the negative terminal.

Drift speed

The mean speed of charge carriers along a conductor.

Electromotive force (emf)

Energy supplied per unit charge by a source, in volts.

Internal resistance

The resistance inside a source, which causes lost volts when current flows.

Kirchhoff's laws

Charge is conserved at a junction. Energy is conserved around a loop.

Lost volts

The p.d. Ir across the internal resistance.

Ohm (Ω)

The unit of resistance, one volt per ampere.

Ohmic conductor

A conductor whose current is proportional to p.d. at constant temperature.

Potential difference (p.d.)

Energy transferred per unit charge between two points.

Potential divider

Two or more resistors in series used to obtain a fraction of a p.d.

Resistivity

A property of a material: RA/L, in ohm metres.

Tangent resistance

The slope dV/dI at a point on a characteristic.

Terminal p.d.

The p.d. across a source's terminals while it supplies current.

Tolerance

The permitted percentage difference between a component's marked and actual value.

Volt (V)

The unit of p.d., one joule per coulomb.

These are general sources for further reading. Titles and publishers are given without links. Look them up through the publisher's home page or your library, and check which edition suits your course.

• University Physics, by Young and Freedman (Pearson).
• Physics for Scientists and Engineers, by Serway and Jewett (Cengage).
• Fundamentals of Physics, by Halliday, Resnick and Walker (Wiley).
• Physics, by Giancoli (Pearson).
• An Introduction to Error Analysis, by John R. Taylor (University Science Books).
• Practical Physics, by Squires (Cambridge University Press).
• Your own examination board's published specification and practical handbook (check the board's home page for the current version).
• Your national standards body or the International Bureau of Weights and Measures (BIPM) for the definitions of the SI units.

This page is an independent learning resource. It is not endorsed by any examination board.