Current, Resistance, and Circuits

Introduction

Electric circuits connect some of the most important ideas in physics: charge, energy, electric fields, materials, and conservation laws. In this aiMOOC for Grades 11–13, you will learn how to describe electric current, model resistance, analyze series and parallel networks, use Ohm's law, apply Kirchhoff's circuit laws, calculate electrical power, and interpret measurements made with meters.

You should already be comfortable rearranging algebraic equations and using SI units. By the end, you should be able to move between a physical circuit, a circuit diagram, equations, measurements, and an explanation of what the charge carriers and energy are doing.

The diagram above distinguishes conventional current from electron motion. In a metallic wire, mobile electrons drift opposite to the conventional-current direction. Circuit analysis normally uses conventional current, defined as the direction positive charge would move.

Learning Goals

After completing this course, you should be able to:

Electric current: Define current as a rate of charge transfer and use $I = \frac{Δ Q}{Δ t}$ .
Electrical resistance: Relate resistance to voltage, current, resistivity, dimensions, and temperature.
Ohm's law: Distinguish an ohmic relationship from a general current-voltage relationship.
Series and parallel circuits: Determine equivalent resistance and predict current and voltage distributions.
Kirchhoff's circuit laws: Use junction and loop rules to analyze multi-loop circuits.
Electric power: Calculate power and energy transfer in resistive devices.
Multimeter: Choose appropriate meter connections and interpret the effect of real measuring instruments.

Electric Current

Charge Flow and Current

Electric current is the rate at which electric charge passes through a chosen cross-section: $I = \frac{Δ Q}{Δ t}$ .

The SI unit is the ampere (A), where $1 A = 1 C s^{- 1}$ . A current of 0.50 A means that, on average, 0.50 coulomb of charge passes the cross-section every second.

For a steady current, $Q = I t$ . For a changing current, the instantaneous current is $I = \frac{d Q}{d t}$ . In metallic conductors, current arises from the drift motion of electrons within a material that already contains mobile charge carriers. The individual drift speed can be very small even though a change in the electric field is established through the circuit much faster.

Conventional Current and Electron Flow

Circuit arrows normally show conventional current. That convention was established before the electron was identified. In a metal, electrons carry negative charge and therefore drift opposite to the conventional-current direction. Both descriptions are consistent if signs are handled correctly.

For a conductor with charge-carrier number density $n$ , carrier charge magnitude $q$ , cross-sectional area $A$ , and drift speed magnitude $v_{d}$ , the current magnitude can be written as: $I = n q A v_{d}$ .

This microscopic relation connects a circuit-scale current to the motion of enormous numbers of charge carriers.

Resistance and Ohm's Law

Resistance as a Circuit Property

Resistance describes how strongly a component opposes current for a given potential difference. For a two-terminal component, $R = \frac{V}{I}$ , with resistance measured in ohms ( $Ω$ ).

A component is ohmic over a range of operating conditions when voltage is proportional to current, so its resistance is approximately constant: $V = I R$ .

Ohm's law is not a universal law for every device under every condition. Diodes, filament lamps over wide temperature ranges, and many semiconductor devices can have nonlinear current-voltage characteristics.

Current-Voltage Characteristics

For an ohmic resistor at constant physical conditions, a graph of current against voltage is a straight line through the origin. If current is on the vertical axis and voltage is on the horizontal axis, the slope is $1 / R$ . If voltage is on the vertical axis and current is on the horizontal axis, the slope is $R$ .

When you interpret an experimental graph, always check the axis order before using a slope to infer resistance.

Resistivity, Geometry, and Temperature

For a uniform conductor of length $L$ and cross-sectional area $A$ , $R = ρ \frac{L}{A}$ , where $ρ$ is the material's resistivity in ohm-metres.

This relation predicts that a longer wire has greater resistance and a thicker wire has lower resistance, provided the material and temperature are unchanged. Resistivity is a material property, while resistance depends on both material and geometry.

For many metals over a moderate temperature range, resistance can be approximated by $R = R_{0} [1 + α (T - T_{0})]$ , where $α$ is the temperature coefficient of resistance. The linear approximation has a limited temperature range and should not be treated as exact at all temperatures.

Reading Real Resistors

Many fixed resistors use coloured bands to encode resistance and tolerance. In practical work, read the band code and then verify the component with a meter when accuracy matters.

Series and Parallel Circuits

Series Connections

Components are in series when they form a single current path with no branching between them. The same current passes through each series component. For resistors, $R_{e q} = R_{1} + R_{2} + \dots + R_{N}$ .

The source voltage is distributed among the series resistors. For ideal wires and resistors, $V_{s o u r c e} = V_{1} + V_{2} + \dots + V_{N}$ .

A larger series resistance receives a larger voltage drop when the same current passes through every resistor.

Parallel Connections

Components are in parallel when they are connected across the same two nodes. The voltage across every parallel branch is the same. The total current entering the combination equals the sum of branch currents.

For parallel resistors, $\frac{1}{R_{e q}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{N}}$ .

The equivalent resistance of a parallel group is smaller than the smallest individual branch resistance because adding a branch provides another path for charge flow.

Mixed Networks and a Reliable Reduction Strategy

For a mixed resistor network, do not decide that components are in series merely because they are drawn next to one another. Trace the nodes. Two components are in parallel only if both of their terminals connect to the same two nodes. Two components are in series only if the same current must pass through both and their shared node has no other branch.

A useful strategy is to identify one reducible series or parallel group, replace it with its equivalent resistance, redraw the circuit, and repeat. After finding total current, work backward through the reductions to recover branch currents and voltage drops.

Kirchhoff's Circuit Laws

Some circuits cannot be reduced entirely by simple series-parallel combinations. Kirchhoff's rules provide a systematic method based on conservation principles. The junction rule expresses conservation of charge, and the loop rule expresses conservation of energy.

Junction Rule

At any node, $\sum I_{i n} = \sum I_{o u t}$ .

No net charge accumulates at an ideal steady-state junction. If you assign a current direction and the algebra gives a negative result, the real current flows opposite to your assumed arrow.

Loop Rule

Around any closed loop, $\sum Δ V = 0$ .

Choose a traversal direction and apply a consistent sign convention. Crossing an ideal source from its negative terminal to its positive terminal is a potential rise. Moving through a resistor in the direction of the assumed current gives a potential drop of $I R$ . The opposite traversal gives the opposite sign.

Solving a Multi-Loop Circuit

A disciplined method reduces sign errors:

Circuit diagram: Label nodes, sources, resistors, and assumed branch-current directions.
Junction equation: Write enough independent junction equations to relate the branch currents.
Loop equation: Choose independent loops and write one consistent voltage equation for each.
Simultaneous equations: Solve the resulting system algebraically.
Dimensional analysis: Check units and substitute the currents back into the original equations.
Physical interpretation: Treat a negative current as evidence that the true direction is opposite to your initial assumption.

Kirchhoff's rules can also analyze simple circuits, but series-parallel reduction is often faster when the topology allows it.

Measuring Current, Voltage, and Resistance

Ammeter and Voltmeter Connections

An ammeter measures current and is connected in series with the branch being measured. An ideal ammeter has zero resistance; a real one has very small resistance.

A voltmeter measures potential difference and is connected in parallel across the component or points of interest. An ideal voltmeter has infinite resistance; a real one has very large resistance.

Connecting a current meter directly across a low-resistance source can create a dangerously large current and can damage the instrument or its fuse.

Practical Measurement and Lab Safety

Before powering a circuit, check the component ratings, meter mode, lead sockets, range, and polarity where relevant. Begin with an appropriate high range if the expected value is uncertain. De-energize the circuit before changing resistance ranges or rewiring components.

For school experiments, use teacher-approved low-voltage power supplies or batteries. Do not experiment with mains electricity. Real sources, wires, meters, and contacts have nonzero internal resistance, so measured values can differ from an ideal calculation.

Electrical Power and Energy

The power transferred by an electrical component is $P = I V$ .

For an ohmic resistor, combining this with Ohm's law gives $P = I^{2} R$ and $P = \frac{V^{2}}{R}$ .

Electrical energy transferred during time $t$ at constant power is $E = P t$ .

These equations help explain why a resistor can heat strongly when current increases. Because $P = I^{2} R$ , doubling current through the same resistance increases the heating power by a factor of four.

Sources, EMF, and Internal Resistance

An ideal voltage source maintains a fixed potential difference independent of current. Real batteries and power supplies have internal effects that can be modeled, over a suitable operating range, by an electromotive force $ℰ$ in series with internal resistance $r$ .

When a source delivers current $I$ , a common model for terminal voltage is $V_{t e r m i n a l} = ℰ - I r$ .

This explains why a battery's terminal voltage can fall under load. The internal power transfer is $I^{2} r$ , so high currents can produce significant heating and reduce useful terminal voltage.

Conceptual Connections

Charge Conservation

The Kirchhoff junction rule follows from conservation of electric charge. In steady state, charge does not continually pile up at a node. Current entering must therefore equal current leaving.

Energy Conservation

The loop rule follows from energy conservation. A charge that completes a closed loop returns to its starting electric potential, so the algebraic sum of potential changes around the loop is zero. Sources transfer energy to charges, while resistors and other loads transfer electrical energy to thermal, light, mechanical, or other forms.

Ideal Models and Real Devices

Circuit theory often begins with ideal wires, ideal meters, ideal voltage sources, and ohmic resistors. These assumptions make the mathematics transparent. Experimental discrepancies then become useful: they can reveal internal resistance, heating, contact resistance, meter loading, tolerance, or non-ohmic behavior.

At Grades 11–13, a strong solution should state which model is being used and whether the model's assumptions are reasonable for the situation.

Interactive Tasks

Quiz: Test Your Knowledge

What does electric current measure? (Rate of charge flow) (!Energy per unit charge) (!Resistance per unit length) (!Power transferred per second)

Which relation describes an ohmic resistor at constant conditions? (Voltage equals current times resistance) (!Current equals voltage times power) (!Resistance equals current times charge) (!Power equals resistance divided by voltage)

What is the SI unit of electrical resistance? (Ohm) (!Ampere) (!Coulomb) (!Watt)

Which quantity is the same through ideal resistors connected in series? (Current) (!Voltage) (!Power) (!Resistance)

Which quantity is the same across ideal branches connected in parallel? (Voltage) (!Current) (!Charge) (!Power)

Which conservation law is directly represented by Kirchhoff's junction rule? (Conservation of charge) (!Conservation of momentum) (!Conservation of mass only) (!Conservation of angular momentum)

Which conservation principle is directly represented by Kirchhoff's loop rule? (Conservation of energy) (!Conservation of momentum) (!Conservation of particle number) (!Conservation of volume)

Which expression gives electrical power for any two-terminal component using current and voltage? (Power equals current times voltage) (!Power equals current divided by voltage) (!Power equals resistance times charge) (!Power equals voltage divided by time)

How should an ammeter be connected to measure branch current? (In series with the branch) (!In parallel across the source) (!Across an open switch) (!Between two unrelated nodes)

At fixed voltage, what happens to current when resistance increases? (Current decreases) (!Current increases) (!Current remains identical) (!Current becomes voltage)

Memory Game

Electric current	Rate of electric charge transfer
Resistance	Opposition to current for a given potential difference
Resistivity	Material property relating resistance to conductor geometry
Ohmic conductor	Device with an approximately linear voltage-current relation under fixed conditions
Series circuit	Connection in which the same current passes through each component
Parallel circuit	Connection in which branches share the same two nodes
Junction rule	Statement that current entering a node equals current leaving it
Loop rule	Statement that potential changes around a closed path sum to zero

Drag and Drop

Match the correct terms.	Topic
Same current through each component	Series connection
Same voltage across each branch	Parallel connection
Current balance at a node	Junction rule
Potential changes sum to zero around a closed path	Loop rule
Measures potential difference across two points	Voltmeter

...

Crossword Puzzle

Current	What quantity is the rate of electric charge flow?
Voltage	What quantity measures electric potential difference?
Resistance	What quantity is measured in ohms?
Resistor	What component is commonly used to limit current?
Junction	What do you call a node where three or more branches meet?
Ammeter	What instrument measures electric current?

LearningApps

Cloze Text

Electric current is the rate of

transfer through a cross-section. The SI unit of current is the

. For an ohmic resistor under fixed conditions, voltage is proportional to

. A uniform conductor's resistance depends on its material resistivity, length, and cross-sectional

. In a series circuit, the same

passes through every component. In a parallel circuit, each branch has the same

between its two nodes. Kirchhoff's junction rule expresses conservation of

. Kirchhoff's loop rule expresses conservation of

. An ammeter is connected in

with the branch whose current is measured. A voltmeter is connected in

across the points whose potential difference is measured.

Open-Ended Tasks

Easy

Current diary: Identify five electrical devices around you and write one sentence for each describing where current flows when the device operates.
Circuit symbol poster: Create a clear poster showing common circuit symbols for a cell, battery, resistor, variable resistor, lamp, switch, ammeter, and voltmeter.
Ohm's law graph: Use a small data set of voltage and current measurements to draw a graph, determine whether the device is approximately ohmic, and estimate its resistance.
Series-parallel photo study: Find or create two safe low-voltage circuit examples and label which parts are in series and which are in parallel.

Standard

Resistance investigation: Carry out a low-voltage experiment to investigate how the resistance of a wire depends on length while keeping material, cross-sectional area, and temperature as controlled as possible.
Multimeter tutorial: Produce a short illustrated guide or video showing how to measure voltage, current, and resistance safely with a digital multimeter.
Power comparison: Compare two resistive loads using measured voltage and current, calculate their power, and explain any difference between predicted and measured heating.
Circuit interview: Interview an electrician, electronics technician, laboratory technician, or physics teacher about common circuit-measurement mistakes and summarize the practical advice.

Advanced

Kirchhoff modeling project: Design a multi-loop resistor circuit, predict every branch current with Kirchhoff's rules, construct a safe low-voltage version, and compare theory with measurement.
Internal resistance experiment: Estimate the internal resistance of a safe battery or laboratory source from terminal-voltage measurements under at least three loads and evaluate uncertainty.
Non-ohmic device study: Investigate the current-voltage characteristic of an LED, filament lamp, or another teacher-approved component and explain why a single constant resistance is not sufficient across the full operating range.
Circuit simulation critique: Build the same circuit in a simulator and in the laboratory, compare ideal and measured values, identify at least three non-ideal effects, and present the results as a technical report or video.

Learning Assessment

Model selection: Given a real circuit description, decide whether series-parallel reduction, Ohm's law, or Kirchhoff's rules are required, and justify the choice.
Error diagnosis: Analyze a circuit in which measured current differs from the ideal prediction and rank plausible causes such as resistor tolerance, internal resistance, heating, meter loading, and connection error.
Energy transfer analysis: For a resistor network with a known source voltage, calculate branch powers and explain how the total electrical power is consistent with energy conservation.
Experimental design: Plan an experiment that distinguishes an ohmic resistor from a non-ohmic device, including controlled variables, measurement ranges, graph choice, and safety precautions.
Complex circuit transfer: Use junction and loop equations to solve an unfamiliar two-loop circuit, then verify the solution by checking charge conservation, energy conservation, and limiting behavior.
Communication task: Explain to a younger student why electrons can drift slowly while a lamp responds quickly after a switch is closed, using a correct field-and-circuit model rather than saying that one electron travels instantly from the battery to the lamp.

Evidence of Learning

Evidence that you have mastered this topic can include knowledge of current, voltage, resistance, resistivity, power, circuit topology, and conservation laws; skills in drawing diagrams, choosing equations, solving simultaneous equations, using SI units, interpreting graphs, and connecting meters correctly; products such as annotated circuit diagrams, experimental data tables, current-voltage graphs, uncertainty analyses, videos, reports, and simulations; and transfer achievements in which you can apply the same principles to unfamiliar networks, troubleshoot discrepancies, evaluate the limits of ideal models, and explain how circuit behavior follows from charge and energy conservation.

A particularly strong portfolio shows agreement between calculation, simulation, and safe measurement while also explaining any remaining differences.