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Current Electricity & Circuits

Q: Why do electrons drift slowly, but the light turns on instantly?

The electrons drift slowly, at about a millimetre per second. The electric field that drives the circuit travels through the wire at close to the speed of light, so all the electrons start moving almost at once.

Q: Why are resistors in parallel better for a circuit with a fixed supply voltage?

Each branch gets the full supply voltage, so adding branches lowers the total resistance and lets more current flow. Components in parallel keep working if one of them fails.

Q: How does an ammeter differ from a voltmeter?

An ammeter is connected in series and has very low resistance, so it does not disturb the current. A voltmeter is connected in parallel and has very high resistance, so it draws almost no current.

Q: Why does a low-resistance wire get hot?

Current through a resistance produces heat at a rate of I squared times R, so a short circuit with very low resistance can draw large currents that overheat.

Q: Do batteries supply constant current?

No. A battery supplies a voltage, and the current depends on the load. Internal resistance means the terminal voltage falls when the current is large.

Ohm, Kirchhoff, resistors, capacitors, RC circuits and measuring instruments. 54 topics, from beginner to advanced. Open the interactive version with labs and quizzes

Current electricity is the flow of charge through conductors. This area defines current, drift velocity, resistance and resistivity, Ohm's law, and how resistors combine in series and parallel. It goes on to Kirchhoff's rules for circuits, the RC charging and discharging behaviour of capacitors, and how ammeters and voltmeters are built and used.

How the topics connect

Current electricity explains how charge flows through circuits. Ohm's law, resistance and power link voltage and current, while Kirchhoff's laws and network theorems let you solve circuits with many branches. Capacitors add time-dependent behaviour through RC circuits, and the area ends with batteries and electrical safety.

Real-world uses

Home wiring and safety

Calculate the current a circuit carries and choose fuses and cable thicknesses that prevent overheating.

Electronics design

Design voltage dividers, LED current limiters and bias networks using Ohm's law and Kirchhoff's rules.

Sensors and measurement

Use a Wheatstone bridge to measure small changes in resistance, as in strain gauges and temperature sensors.

Battery and power planning

Estimate how long a battery lasts and how much power a device draws, from its voltage and resistance.

Home electrics

Electricians use Ohm's law and the power formula to size cables, fuses and circuit breakers for each appliance.

Electronics design

Designers use RC timing, diodes and transistors to build timers, power supplies and digital logic.

Beginner

Electric current

Rate of flow of charge.

Key relationI = dQ/dt

Current is the rate of flow of charge past a point: I = dQ/dt, in amperes (1 A = 1 C/s). In metals it is carried by electrons; in electrolytes by ions.

Worked example

A phone charger at 2 A moves 2 C each second.

Common mistake

Current is a scalar with a conventional direction.

Beginner

Direction of current

Conventional current flows from positive to negative.

Conventional current flows from the positive terminal to the negative, the way positive charges would move. Electrons actually drift the opposite way in wires.

Worked example

Arrows on circuit diagrams follow the convention.

Common mistake

The choice doesn't change physics, only signs.

Intermediate

Drift velocity

Slow average speed of charge carriers.

Key relationI = nAev_d

Electrons in a wire drift slowly: I = nAev_d. In copper v_d is about 0.1 mm/s even for amps of current, yet the signal travels near the speed of light because the field is established everywhere almost at once.

Worked example

1 A in 1 mm² copper: v_d ≈ 0.074 mm/s.

Common mistake

The lamp lights instantly because charges are already present throughout the wire.

Beginner

Ohm's law

Voltage is proportional to current for ohmic conductors.

Key relationV = IR

For many conductors the current is proportional to voltage: V = IR with constant R at fixed temperature. Materials obeying this are ohmic.

Worked example

12 V across 4 Ω: 3 A.

Common mistake

Diodes, filament lamps and thermistors are not ohmic.

Beginner

Resistance

Opposition to current.

Key relationR = ρL/A

R = ρL/A measures opposition to current: longer and thinner wires resist more. Unit: ohm (V/A).

Worked example

Copper wire of 10 m and 1 mm²: 0.17 Ω.

Common mistake

Resistance depends on material and shape.

Intermediate

Resistivity

A material property independent of shape.

Key relationρ = RA/L

ρ is a property of the material: copper 1.7×10⁻⁸ Ω·m, nichrome 1.1×10⁻⁶, glass 10¹⁰ or more. It's independent of the wire's size.

Worked example

Heating elements use high-ρ alloys.

Common mistake

Don't mix resistance and resistivity.

Intermediate

Conductivity

Reciprocal of resistivity.

Key relationσ = 1/ρ

σ = 1/ρ in S/m measures how readily a material conducts. Metals have high σ, insulators very low.

Worked example

Silver 6.3×10⁷ S/m.

Common mistake

Used in the Drude model.

Intermediate

Temperature dependence of resistance

Metals' resistance rises with temperature.

Key relationR = R₀(1+αΔT)

Metal resistance rises with temperature: R = R₀(1 + αΔT), because lattice vibrations scatter electrons more. Semiconductors' resistance falls with temperature.

Worked example

Tungsten filament at 2500 °C has about 12 times the cold resistance.

Common mistake

Cold start-up currents are therefore large.

Advanced

Superconductors

Materials with zero resistance below a critical temperature.

Below a critical temperature some materials have exactly zero resistance and expel magnetic fields (Meissner effect). Used in MRI magnets and maglev trains.

Worked example

Mercury at 4.2 K; cuprates up to 138 K.

Common mistake

Not just very low resistance but zero.

Intermediate

Semiconductors and thermistors

Resistance falls as temperature rises.

Semiconductors (silicon, germanium) have few carriers at low temperature; heating releases more, so resistance drops. NTC thermistors exploit this for temperature sensing.

Worked example

Thermistors in digital thermometers.

Common mistake

Doping controls conductivity.

Beginner

Resistors in series

Resistances add.

Key relationR = R₁+R₂

The same current flows through each, voltages add, and R = R₁ + R₂ + ... The largest resistor gets the largest voltage.

Worked example

10 Ω + 20 Ω = 30 Ω.

Common mistake

One break stops the entire circuit.

Beginner

Resistors in parallel

Reciprocals add.

Key relation1/R = 1/R₁+1/R₂

Each gets the full voltage, currents add and 1/R = 1/R₁ + 1/R₂. The total is smaller than the smallest branch.

Worked example

10 Ω ∥ 10 Ω = 5 Ω.

Common mistake

Adding a branch always reduces the total resistance.

Beginner

Colour code of resistors

Bands encode value and tolerance.

Bands encode digits, multiplier and tolerance: black 0, brown 1, red 2, orange 3, yellow 4, green 5, blue 6, violet 7, grey 8, white 9. Red-red-brown-gold = 22 × 10 = 220 Ω ±5%.

Worked example

Yellow-violet-orange = 47 kΩ.

Common mistake

Read from the end opposite the tolerance band.

Beginner

Electric power

Rate of energy use.

Key relationP = VI = I²R = V²/R

P = VI = I²R = V²/R. Energy used E = Pt. A 2 kW heater on 230 V draws 8.7 A.

Worked example

A 60 W bulb at 120 V has R = 240 Ω.

Common mistake

Use the form matching the known quantities.

Beginner

Heating effect of current

Joule heating.

Key relationH = I²Rt

Collisions of charge carriers with the lattice convert electrical energy to heat: H = I²Rt (Joule heating). Used in heaters and fuses; wasted in cables.

Worked example

A 10 A current through a 2 Ω element for 60 s: 12 kJ.

Common mistake

Doubling current quadruples heating.

Beginner

Electrical energy and cost

Bills use kilowatt-hours.

Domestic meters record kWh. Cost = kW × hours × tariff.

Worked example

A 100 W TV for 5 h is 0.5 kWh.

Common mistake

Convert W to kW first.

Intermediate

EMF and internal resistance

Terminal voltage drops under load.

Key relationV = ε − Ir

A real source has EMF ε and internal resistance r, so terminal voltage V = ε − Ir drops under load. EMF is the open-circuit voltage.

Worked example

A 12 V battery with r = 0.1 Ω delivering 30 A: V = 9 V.

Common mistake

Short circuit current is ε/r.

Intermediate

Cells in series and parallel

Series adds emf; parallel reduces internal resistance.

In series EMFs add (and internal resistances add): more voltage. In parallel EMF stays the same but the internal resistance falls and capacity increases.

Worked example

Four 1.5 V cells in series: 6 V.

Common mistake

Mix only identical cells in parallel.

Intermediate

Kirchhoff's current law

Currents into a junction sum to zero.

Key relationΣI = 0

At any junction, the sum of currents in equals the sum out: ΣI = 0, from conservation of charge.

Worked example

3 A and 2 A in give 5 A out.

Common mistake

Assign directions, and let signs sort themselves out.

Intermediate

Kirchhoff's voltage law

Voltages around a loop sum to zero.

Key relationΣV = 0

Around any closed loop the sum of EMFs equals the sum of potential drops: ΣV = 0, from energy conservation.

Worked example

A 12 V battery with two resistors: V₁ + V₂ = 12.

Common mistake

Be careful with signs: drops along the current direction.

Intermediate

Wheatstone bridge

Balanced when ratios of resistances are equal.

Key relationP/Q = R/S

Four resistors in a diamond with a galvanometer across the middle. When P/Q = R/S no current flows through it, which lets you find an unknown resistance precisely.

Worked example

P = 100, Q = 200, R = 150: S = 300 Ω.

Common mistake

Works because of equal potentials, regardless of the supply voltage.

Intermediate

Meter bridge

Practical Wheatstone bridge using a wire.

A practical Wheatstone bridge using a 1 m wire and a sliding contact; the balance length ratio gives the resistance ratio: X = R(100 − l)/l.

Worked example

R = 5 Ω, l = 40 cm: X = 7.5 Ω.

Common mistake

Use the middle of the wire for accuracy.

Intermediate

Potentiometer

Compares emfs without drawing current.

A uniform wire with steady current compares emfs by balancing potential drops along the wire without drawing current from the cell, so it measures true EMF.

Worked example

ε₁/ε₂ = l₁/l₂.

Common mistake

More accurate than a voltmeter.

Intermediate

Ammeter and voltmeter

Shunts and series resistors extend galvanometer range.

An ammeter is connected in series and should have very low resistance (a galvanometer with a shunt in parallel). A voltmeter is connected in parallel and should have very high resistance (a galvanometer with a series resistor).

Worked example

Shunt R_s = I_gG/(I − I_g).

Common mistake

Wrong connection can burn an ammeter.

Intermediate

Galvanometer

A coil in a magnetic field deflects with current.

A coil in a magnetic field twists against a spring in proportion to the current, moving a pointer. It is the base of analogue meters.

Worked example

Typically full-scale 50 μA.

Common mistake

Sensitive and delicate.

Beginner

Multimeter

Combined instrument for V, I and R.

One instrument to measure V (AC/DC), I and R using switches that choose the shunt or series resistance; digital versions use ADCs.

Worked example

Check the range and the lead position (A vs V) before connecting.

Common mistake

Measure resistance only with power off.

Advanced

Maximum power transfer

Load equal to source resistance gives maximum power.

Key relationR_L = r

A load draws the most power from a source when its resistance equals the source's internal resistance: R_L = r, giving P_max = ε²/4r with only 50% efficiency.

Worked example

ε = 12 V, r = 2 Ω: 18 W.

Common mistake

Power stations aim for high efficiency instead.

Advanced

Thevenin's theorem

Any linear network is equivalent to a voltage source and a resistor.

Any linear two-terminal network is equivalent to a voltage source V_th (open-circuit voltage) in series with a resistance R_th (the resistance seen with sources turned off).

Worked example

Simplifies analysing the effect of different loads.

Common mistake

Replace voltage sources by shorts and current sources by opens.

Advanced

Norton's theorem

Equivalent current source with parallel resistor.

Equivalent to a current source I_N (short-circuit current) in parallel with resistance R_N = R_th. Related to Thevenin by source transformation.

Worked example

V_th = I_N R_N.

Common mistake

Check which is easier.

Advanced

Superposition theorem

Response is sum of responses from each source alone.

In a linear circuit the response is the sum of the responses to each independent source acting alone with the others zeroed.

Worked example

Two batteries in a network: find each contribution and add.

Common mistake

Not applicable to power calculations directly.

Advanced

Delta–star conversion

Convert between triangle and star resistor networks.

A triangle (Δ) of three resistors can be replaced by a star (Y) with R_a = R₁R₂/(R₁ + R₂ + R₃) etc., simplifying bridge circuits.

Worked example

Three 30 Ω Δ → three 10 Ω Y.

Common mistake

Both forms give the same terminal behaviour.

Intermediate

RC circuit charging

Capacitor voltage rises exponentially.

Key relationV = V₀(1−e^{−t/RC})

After closing the switch a capacitor charges as V_C = V₀(1 − e^{−t/RC}); the current decays as I = (V₀/R)e^{−t/RC}.

Worked example

R = 10 kΩ, C = 100 μF: τ = 1 s; after 1 s V_C = 63%.

Common mistake

Capacitor never fully charges in finite time.

Intermediate

RC circuit discharging

Capacitor voltage decays exponentially.

Key relationV = V₀e^{−t/RC}

A charged capacitor discharges through a resistor: V = V₀e^{−t/RC}. After 5RC it has fallen to 0.7%.

Worked example

Camera flash discharge.

Common mistake

Current direction reverses compared to charging.

Intermediate

Time constant

Time for the capacitor to reach about 63% of final voltage.

Key relationτ = RC

τ = RC (seconds) is the time for the voltage to change by 63% of its final change. Larger R or C means slower change.

Worked example

1 MΩ × 1 μF = 1 s.

Common mistake

After one τ it's not finished; wait 5τ.

Intermediate

Current in RC circuit

Charging current decays exponentially.

Key relationI = (V/R)e^{−t/RC}

Initially the full V₀/R flows (capacitor acts like a short), decaying to zero as the capacitor charges and becomes an open circuit.

Worked example

At t = τ I = 0.37 I₀.

Common mistake

Different from the capacitor voltage curve.

Intermediate

Capacitor networks

Combine series and parallel capacitors.

Reduce step by step using series and parallel rules. For steady DC the capacitors carry no current.

Worked example

Two in parallel (2 + 3 = 5 μF) in series with 5 μF → 2.5 μF.

Common mistake

Careful: series is reciprocal addition for capacitors.

Intermediate

Diodes

Components that conduct in one direction.

A diode conducts in one direction (forward biased, above about 0.7 V for silicon) and blocks in the other. LEDs emit light when forward biased.

Worked example

A diode in series with 5 V and 1 kΩ passes 4.3 mA.

Common mistake

Reverse breakdown can destroy it.

Advanced

Rectifier circuits

Convert AC to DC using diodes.

Diodes convert AC to DC: half-wave uses one diode, full-wave uses a bridge of four; a smoothing capacitor reduces ripple.

Worked example

Phone chargers.

Common mistake

Output isn't pure DC without smoothing.

Advanced

Transistor as a switch and amplifier

Small base current controls a large collector current.

A small base current in a bipolar transistor controls a larger collector current (I_C = βI_B). Used as a switch (saturated/off) or as a linear amplifier.

Worked example

β = 100: 0.1 mA controls 10 mA.

Common mistake

Needs a proper bias.

Intermediate

Logic gates

AND, OR, NOT built from transistors.

AND, OR, NOT, NAND, NOR, XOR built from transistors process binary signals. Combinations make adders, memory and computers.

Worked example

AND outputs 1 only if both inputs are 1.

Common mistake

NAND is universal.

Beginner

Electrical safety

Fuses, earthing and circuit breakers prevent shock and fire.

Currents above ~10 mA through the body can cause muscle lock, 100 mA can be fatal. Fuses, circuit breakers, earth wires and RCDs protect.

Worked example

Fuse rated slightly above normal current.

Common mistake

Water lowers body resistance.

Beginner

Household wiring

Live, neutral and earth wires.

Live (230 V or 120 V), neutral and earth wires; appliances are in parallel; switches and fuses go in the live wire. Earth gives a low-resistance path for faults, blowing the fuse.

Worked example

Plug fuse 3, 5 or 13 A.

Common mistake

Switch in the neutral leaves the appliance live.

Intermediate

Electrolysis

Current drives chemical changes.

Key relationm = ZIt

Current through an electrolyte moves ions to the electrodes, depositing or releasing substances. Used for plating and refining metals and making aluminium.

Worked example

Copper plating.

Common mistake

Mass deposited ∝ charge passed.

Intermediate

Faraday's laws of electrolysis

Mass deposited is proportional to charge.

Mass deposited m = ZIt = (M/nF)It; one faraday (96,485 C) deposits one mole of charge equivalent.

Worked example

Copper (M = 63.5, n = 2): 1 A for 1 h deposits 1.19 g.

Common mistake

Z is electrochemical equivalent.

Advanced

Thermoelectric effects

Seebeck, Peltier and Thomson effects convert heat and electricity.

Seebeck: a temperature difference across two metals generates voltage (thermocouples). Peltier: current moves heat (cooling). Thomson: heating along a wire with a gradient.

Worked example

K-type thermocouple gives 41 μV/K.

Common mistake

Reversible effects, unlike Joule heating.

Beginner

Batteries

Chemical cells that supply current.

Chemical energy converts directly into electrical energy through redox reactions at two electrodes. Primary cells are used once, secondary cells recharge.

Worked example

Li-ion cells have 3.7 V and about 250 Wh/kg.

Common mistake

Capacity in Ah and energy in Wh differ.

Intermediate

Charging and discharging batteries

Capacity, internal resistance and efficiency.

Capacity (Ah) is how long a battery can supply current; real capacity is lower at high currents. Charging needs a voltage above the battery's EMF; efficiency is <100%.

Worked example

2000 mAh at 100 mA lasts about 20 h.

Common mistake

Deep discharging shortens lifetime.

Beginner

Electric circuits symbols

Standard diagrams for components.

Standard symbols: cell, battery, resistor, variable resistor, lamp, switch, capacitor, ammeter (A), voltmeter (V), diode. Circuit diagrams show the connections, not the physical layout.

Worked example

Learn them to read and draw circuits.

Common mistake

Ammeter in series, voltmeter in parallel.

Beginner

Short circuit

Low-resistance path causing huge current.

A near-zero resistance path causes a huge current I = ε/r, heating wires and perhaps starting fires. Fuses and breakers disconnect it.

Worked example

12 V battery with r = 0.01 Ω gives 1200 A.

Common mistake

A short is a low resistance, not a broken circuit.

Beginner

Series vs parallel in homes

Appliances connect in parallel to get full voltage.

Appliances in parallel each get the full mains voltage and work independently. Christmas lights in series all go out if one fails.

Worked example

A kettle and a toaster on the same circuit draw the sum of their currents.

Common mistake

Too many appliances can overload the circuit.

Advanced

Drude model

Electrons as a gas colliding with ions give Ohm's law.

Key relationσ = ne²τ/m

Treats conduction electrons as a gas accelerated by the field and scattered after a mean time τ: σ = ne²τ/m, giving Ohm's law. It also explains the Wiedemann–Franz law approximately.

Worked example

Copper τ ≈ 2.5×10⁻¹⁴ s.

Common mistake

Classical; correct treatment needs quantum mechanics.

Advanced

Hall effect

Voltage perpendicular to current in a magnetic field.

Key relationV_H = IB/nqt

A current-carrying conductor in a perpendicular magnetic field develops a transverse voltage V_H = IB/(nqt). It reveals the sign and density of charge carriers and is used in magnetic sensors.

Worked example

In silicon the sign shows n or p type.

Common mistake

Voltage is small in metals.

Advanced

Noise in circuits

Thermal Johnson noise and shot noise limit measurements.

Thermal (Johnson) noise V² = 4kTRΔf and shot noise limit the smallest detectable signals. Cooling and narrow bandwidth reduce them.

Worked example

1 MΩ at 300 K, 10 kHz: 13 μV rms.

Common mistake

Amplifiers add their own noise.

Advanced

Network analysis with nodes and meshes

Systematic methods solve large circuits.

Mesh analysis writes KVL for each loop; nodal analysis writes KCL at each node and solves linear equations. Matrix methods handle large circuits and computer simulation (SPICE).

Worked example

Two-loop circuit gives two equations.

Common mistake

Pick the method with fewer unknowns.

Frequently asked questions

Why do electrons drift slowly, but the light turns on instantly?

The electrons drift slowly, at about a millimetre per second. The electric field that drives the circuit travels through the wire at close to the speed of light, so all the electrons start moving almost at once.

Why are resistors in parallel better for a circuit with a fixed supply voltage?

Each branch gets the full supply voltage, so adding branches lowers the total resistance and lets more current flow. Components in parallel keep working if one of them fails.

How does an ammeter differ from a voltmeter?

An ammeter is connected in series and has very low resistance, so it does not disturb the current. A voltmeter is connected in parallel and has very high resistance, so it draws almost no current.

Why does a low-resistance wire get hot?

Current through a resistance produces heat at a rate of I squared times R, so a short circuit with very low resistance can draw large currents that overheat.

Do batteries supply constant current?

No. A battery supplies a voltage, and the current depends on the load. Internal resistance means the terminal voltage falls when the current is large.

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