Simple Circuits
A circuit is a network of electrical elements connected into one or more paths. Charge can sustain current only through a closed conducting loop. An open circuit interrupts a path; an ideal short creates a bypass path with essentially zero potential difference. Circuit behavior is determined by connectivity—nodes, branches, and loops—not by how the drawing is stretched or rotated. Standard schematics and conventional-current arrows make that topology explicit.
Topic sequence · 2 of 8
Learning objectives
Objective 1
Identify loops, branches, shared elements, open paths, closed paths, and short-circuit paths from a physical circuit or schematic.
Objective 2
Redraw a circuit while preserving node connectivity and use standard symbols for batteries, resistors, bulbs, switches, capacitors, inductors, ammeters, and voltmeters.
Objective 3
Predict qualitative circuit behavior from topology before applying resistance, power, Kirchhoff-rule, or transient equations from later topics.
- Topic 11.1 conventional current and charge-flow direction.
- Topic 9.2 electric potential difference.
- Topic 10.3 capacitors as circuit elements that can store separated charge.
Definitions first
Representations you must read and create
Representation 1
A physical breadboard circuit and a simplified schematic that have the same nodes even though the wires are drawn very differently.
Representation 2
Node-labeled schematic showing which elements are genuinely in parallel because they connect to the same two nodes.
Representation 3
Switch-state diagrams comparing an open branch, a normal closed load path, and an ideal-wire bypass that shorts the load.
Representation 4
Standard symbols for battery, bulb, switch, capacitor, resistor, ammeter, voltmeter, and inductor, with conventional-current arrows.
Core relationships and calculus
Variables: topological condition
Use: a source plus a complete conducting loop can support continuous charge flow
Variables: I A
Use: ideal open-switch or broken-wire model for that branch
Variables: ΔV V
Use: ideal-wire bypass; do not infer a finite current without the rest of the circuit model
Variables: ΔV V
Use: topological definition of parallel connection
Variables: I A
Use: charge does not accumulate indefinitely between elements in one unbranched path
Variables: ΔV V; I A
Use: meter-placement model used when interpreting schematics
Topic 11.2 is primarily a representation and topology topic, but calculus reappears immediately in later circuit models. The important mathematical preparation is to identify the independent paths and node connections correctly before writing equations. A topologically incorrect schematic produces incorrect integrals, differential equations, and Kirchhoff relations no matter how flawless the algebra is.
Two useful topology constraints come from basic definitions. If two elements connect to the same two ideal-wire nodes, the potential at each shared endpoint is the same, so both elements have the same potential difference: this is what “parallel” means. In a single unbranched steady path, charge cannot accumulate indefinitely between successive elements; therefore the same current passes each element. Formal multi-loop current and potential equations are developed later with Kirchhoff’s rules.
This topic establishes circuit topology and schematic literacy. Do not jump ahead by assuming resistor equivalent-resistance formulas, full Kirchhoff loop/junction equations, or RC/LR/LC transient solutions before their dedicated topics. An “ideal short” means zero potential difference across the short path; it does not by itself mean an actually infinite current, because real sources and wires have nonzero limitations.
Physical examples
Example 1
Opening a switch in the only loop prevents sustained current everywhere in that loop even though a potential difference may remain across the open switch.
Example 2
Two resistors drawn side by side are not necessarily parallel; they are parallel only if both terminals of one connect to the same two nodes as the other.
Example 3
An ideal wire placed directly across a lamp creates a short-circuit bypass. In the ideal model the lamp’s terminal potential difference becomes zero.
Example 4
A circuit element can belong to more than one loop, so a visual “one loop per component” assumption can fail in compound networks.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Redraw-by-nodes method: label nodes first, then redraw the circuit cleanly without changing which terminals share a node.
Reasoning pattern 2
Open-versus-short distinction: an open path blocks current; a short provides a bypass with essentially zero potential difference.
Reasoning pattern 3
Series/parallel test: use connectivity, not visual orientation or closeness on the page.
Reasoning pattern 4
Meter placement: an ideal ammeter belongs in the branch whose current is measured; a voltmeter compares two node potentials and is placed across the element.
Reasoning pattern 5
Conventional-current rule: unless stated otherwise, use conventional current when interpreting AP circuit schematics.
Deeper calculus and limiting-case check
Circuit topology is invariant under continuous redrawing: wires may be bent, stretched, or rearranged without changing the physics as long as node connections remain unchanged. This is the same mathematical idea used in network or graph representations, where connectivity—not geometric distance on the page—defines the structure.
Interactive open / closed / short path check
Toggle the idealized switch states. The model classifies the source-to-source path and whether an ideal wire bypass shorts the load.
Experimental connection
Build a low-voltage battery–lamp circuit with a main switch and a removable bypass wire. Before each change, draw the schematic and predict whether the lamp is off, normally powered, or bypassed. Add an ammeter in series and a voltmeter across the lamp to test meter placement. Record observations, then redraw the same physical circuit in two visually different but topologically equivalent schematics.
Connections across Physics C
Connection 1
Topic 11.1 supplies current direction and charge-flow meaning for every arrow drawn on a circuit schematic.
Connection 2
Topic 9.2 supplies potential difference, which is compared across elements and nodes.
Connection 3
Topic 10.3 introduces capacitors as elements that may appear in circuit loops.
Connection 4
Topics 11.3–11.8 add resistance, power, multi-loop rules, and time-dependent capacitor behavior after the topology is established.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Calling elements parallel because they are drawn next to each other instead of checking whether they share the same two nodes.
Avoid this
Assuming an open switch must have zero potential difference because its current is zero.
Avoid this
Saying a short circuit automatically has “infinite current” instead of recognizing that the ideal short has ΔV=0 and actual current depends on the rest of the model.
Avoid this
Placing an ideal voltmeter in series or an ideal ammeter directly across a source/load.
Avoid this
Treating wire crossings as junctions without checking whether the schematic indicates an actual electrical connection.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What makes a circuit closed?
There must be a complete conducting path through which charge can circulate. A break in the relevant path makes it open.
What exactly is a short circuit?
It is a very-low-resistance bypass path with essentially no potential difference across that path, often bypassing an intended load.
Can an open switch have a voltage across it?
Yes. Zero current through the open gap does not require zero potential difference across the gap.
How do I know whether two elements are parallel?
They are parallel only when both terminals connect to the same two nodes.
Why can I redraw a circuit?
The circuit behavior depends on electrical connectivity. Bending or stretching ideal wires on the page does not change which terminals share a node.