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doPhysics · Electromagnetism 1

Circuits: Kirchhoff's Laws, Networks and RC Transients

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Why this chapter
Why this chapter exists

A single resistor on a single cell is rare. A house, a phone charger or a car is a network of branches that split and join. This chapter gives two rules that solve any such network: charge is conserved at every junction, and energy is conserved round every loop. Before them come the shortcuts, series, parallel, the balanced bridge and symmetry, that make most networks easy. The last two lessons add a capacitor, and the currents stop being steady. They die away with a time constant, the timing behind every camera flash.

You will learn
By the end of this chapter you will
  1. Reduce networks of resistors by series and parallel steps, by the balanced bridge and by symmetry, including the infinite ladder and the cube.
  2. State Kirchhoff's junction and loop rules, the conservation laws behind them, and the sign of every term of a loop equation.
  3. Solve networks with one or two unknown junction potentials by the nodal method, and say which cell is being charged.
  4. Derive and use the charging and discharging laws of an RC circuit and its time constant. Say what the circuit does at \(t = 0\) and after a long time.
  5. Account for the energy of charging and discharging, and read R, C and the emf from an RC graph.
Before you begin
Before you begin

From the chapter Electric Current and Resistance you should know Ohm's law, \(V = IR\), the resistance of a wire, \(\rho L/A\), and the power \(I^2R\). From the chapter Capacitance and Dielectrics you should know \(q = CV\) and the energy \(\tfrac12 CV^2\) of a capacitor. The chapter Electric Potential, Energy and the Dipole gives potential as work per coulomb. Lessons 5 and 6 use the exponential function and one simple integral.

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Lessons

1
Resistors in Series and in Parallel
Resistors in series carry one current and share the voltage; resistors in parallel share one voltage and add their currents. Mixed networks reduce step by step, and a wire cut, folded or stretched is a combination in disguise.
Resistors in seriesResistors in parallelMixed networks of resistorsWires cut folded and stretched
14 min0%
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2
Symmetry, Bridges and Ladders
Networks that are neither series nor parallel. The balanced Wheatstone bridge, rings and polygons of wire, the infinite ladder that contains itself, and the cube solved by symmetry.
The wheatstone bridgeRings and polygons of wireThe infinite ladderThe cube of resistors
16 min0%
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3
Kirchhoff's Laws
The junction rule, from the conservation of charge, and the loop rule, from the conservation of energy. One sign convention for walking round a loop, one loop with two cells, and the potentials round a loop with its zero chosen by earthing.
The junction ruleThe loop rule and its signsOne loop with two cellsPotentials round a loop and earthing
14 min0%
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4
Solving Networks: the Nodal Method
Choose a zero, name the unknown potentials, write every branch current from them and apply the junction rule. One unknown settles two cells and a load, shows which cell is being charged, and gives the voltage of cells in parallel.
The nodal methodTwo cells and a loadA second network by nodesCells in parallel by the junction rule
15 min0%
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5
Charging a Capacitor: the RC Circuit
A capacitor charged through a resistor fills along an exponential with the time constant RC. The loop equation and its solution, what happens just after the switch closes and after a long time, and where the battery's energy goes.
The time constantCharge and current while chargingAt t zero and after a long timeEnergy accounting of charging
15 min0%
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6
Discharging and Reading RC Graphs
A charged capacitor discharging through a resistor loses its charge exponentially, with a current that flows the other way. Halving times like a decay, reading R, C and the emf from a graph, and the heat shared by two resistors.
Discharging a capacitorReading rc graphsHeat shared between two resistors
13 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

balanced
Said of a Wheatstone bridge whose middle points are at one potential, so that the middle branch carries no current; it needs P/Q = R/S.
branch
A stretch of circuit between two junctions; one current flows all along it.
earthing
Joining a point of a circuit to the Earth, which fixes its potential as zero; a single earth wire carries no current.
emf
The work a cell does on each coulomb it drives round a circuit, written \(\varepsilon\); the potential difference between its terminals when no current flows.
equivalent resistance
The single resistance that would draw the same current from the same potential difference as a whole network between two terminals.
infinite ladder
A chain of identical sections of resistors that never ends; removing its first section leaves the same ladder.
internal resistance
The resistance of the material inside a cell, drawn as a small resistor in series with its emf.
junction
A point of a circuit where three or more wires meet, so that the current divides or joins.
Kirchhoff's laws
Two rules for circuits. The current into a junction equals the current out, and the potential changes round a closed loop add to zero.
loop
Any closed path round a circuit that returns to its starting point.
nodal method
Solving a network with the potentials of its junctions as the unknowns. One junction is taken as zero, and the junction rule is applied at the others.
node
Another name for a junction, used especially in the nodal method.
parallel
Two or more components joined between the same two junctions, so that the same potential difference is across all of them.
RC circuit
A circuit in which a capacitor charges or discharges through a resistance.
series
Two or more components joined end to end with no junction between them, so that the same current flows through all of them.
time constant
The product \(\tau = RC\) of an RC circuit. In one time constant a charging capacitor reaches 63 % of its final charge, and a discharging one falls to 37 %.
transient
A current or voltage that changes for a while after a switch is closed or opened and then settles to a steady value.
voltage divider
Resistors in series used to take off a fixed fraction of a voltage; each takes a share in proportion to its resistance.
Wheatstone bridge
Four resistors joined in a diamond, with a cell across one diagonal and a galvanometer or a fifth resistor across the other.