DC & AC circuits

Build a circuit from cells, resistors, lamps, capacitors, inductors and switches, and measure it with meters.

Circuits — Current, potential difference and resistance, in direct and alternating supplies.

Open it in The Lab The equations behind it

Experiments already set up on it

Ohm’s law
One resistor across a cell. Current against voltage is a straight line.
- V = IR, and the gradient of I against V is 1/R
- A resistor that obeys Ohm’s law gives a straight line through the origin
Resistors in series
Two resistors and a switch. The same current flows through both.
- Series resistances add: R = R₁ + R₂
- The same current flows everywhere in a series circuit
- The voltages across the parts add up to the supply voltage
Resistors in parallel
Two resistors side by side. Each gets the full voltage.
- Parallel resistances combine as reciprocals
- Adding a parallel branch increases the total current
- Each branch has the full supply voltage across it
Potential divider
Two resistors sharing 9 V. The output follows R₂/(R₁+R₂).
- The divider rule V_out = V·R₂/(R₁+R₂)
- Why a divider is the usual way to get a smaller voltage
Internal resistance of a cell
The terminal p.d. drops as more current is drawn: V = E − Ir.
- V = E − Ir, so a graph of V against I has gradient −r and intercept E
- Why a battery gets "weaker" under load
Charging a capacitor
Watch the voltage climb towards the supply with time constant RC.
- Exponential charging: V = V₀(1 − e^(−t/RC))
- The time constant τ = RC, and 63 % of the way in one τ
- Current falls as the capacitor fills
Time constant against resistance
Sweep R and see the charging time grow in proportion.
- τ = RC is proportional to R
- Reading a time constant off a graph
AC through an RLC circuit
An AC supply through a resistor, an inductor and a capacitor in series.
- Reactance depends on frequency: X_L = 2πfL and X_C = 1/(2πfC)
- At resonance the two cancel and the current is largest
- Resonant frequency f₀ = 1/(2π√(LC))
I–V curve of a filament lamp
A lamp is not a resistor: as it heats up its resistance climbs, so the graph bends.
- A filament lamp does not obey Ohm’s law — I against V curves away from a straight line
- Resistance rises with temperature, and temperature rises with the power dissipated
- A gradient taken from a curve only means anything at the point it was taken
Build your own circuit
Place components anywhere and draw the wires yourself. Each wire has a material and a thickness, so the leads have resistance like real ones.
- A junction is wherever two ends meet
- Wires have resistance too: R = ρL/A
- Why connecting leads are thick copper and a heating element is thin nichrome

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)
Current from the supplyI (A)
Terminal p.d. of the supplyV (V)
Resistance of the whole circuitR_{total} (Ω)
Power delivered by the supplyP (W)
Voltage across E1V_{E1} (V)
Current through E1I_{E1} (A)
Power in E1P_{E1} (W)
Voltage across R1V_{R1} (V)
Current through R1I_{R1} (A)
Power in R1P_{R1} (W)
Potential at node 1V_{1} (V)

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

E.m.f. of the cellE (V)
Internal resistance of the cellr (Ω)
Peak voltage of the AC supplyV₀ (V)
Frequency of the AC supplyf (Hz)
Resistance R₁R₁ (Ω)
Resistance R₂R₂ (Ω)
Resistance R₃R₃ (Ω)
Capacitance C₁C₁ (F)
Inductance L₁L₁ (H)

The equations behind it

What dc & ac circuits is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Electric current and circuits

$I = \frac{Δ q}{Δ t}$
Current
$V = I R$
Ohm’s law
$R = \frac{ρ L}{A}$
Resistance of a wire — ρ is the resistivity.
$P = V I = I^{2} R = \frac{V ^{2}}{R}$
Power dissipated
$R_{series} = R_{1} + R_{2} + \dots$
Resistors in series
$\frac{1}{R _{parallel}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$
Resistors in parallel
$ε = I (R + r)$
EMF and internal resistance
$I = n A v q$
Current from drift speed

Capacitance

$C = \frac{q}{V}$
Capacitance
$C = \frac{ε _{0} ε _{r} A}{d}$
Parallel plate capacitor
$E = \frac{1}{2} C V^{2}$
Energy stored
$τ = R C$
Time constant
$q = q_{0} e^{- t / R C}$
Discharging a capacitor

Experiments already set up on it

Ohm’s law

One resistor across a cell. Current against voltage is a straight line.

V = IR, and the gradient of I against V is 1/R
A resistor that obeys Ohm’s law gives a straight line through the origin

Resistors in series

Two resistors and a switch. The same current flows through both.

Series resistances add: R = R₁ + R₂
The same current flows everywhere in a series circuit
The voltages across the parts add up to the supply voltage

Resistors in parallel

Two resistors side by side. Each gets the full voltage.

Parallel resistances combine as reciprocals
Adding a parallel branch increases the total current
Each branch has the full supply voltage across it

Potential divider

Two resistors sharing 9 V. The output follows R₂/(R₁+R₂).

The divider rule V_out = V·R₂/(R₁+R₂)
Why a divider is the usual way to get a smaller voltage

Internal resistance of a cell

The terminal p.d. drops as more current is drawn: V = E − Ir.

V = E − Ir, so a graph of V against I has gradient −r and intercept E
Why a battery gets "weaker" under load

Charging a capacitor

Watch the voltage climb towards the supply with time constant RC.

Exponential charging: V = V₀(1 − e^(−t/RC))
The time constant τ = RC, and 63 % of the way in one τ
Current falls as the capacitor fills

Time constant against resistance

Sweep R and see the charging time grow in proportion.

τ = RC is proportional to R
Reading a time constant off a graph

AC through an RLC circuit

An AC supply through a resistor, an inductor and a capacitor in series.

Reactance depends on frequency: X_L = 2πfL and X_C = 1/(2πfC)
At resonance the two cancel and the current is largest
Resonant frequency f₀ = 1/(2π√(LC))

I–V curve of a filament lamp

A lamp is not a resistor: as it heats up its resistance climbs, so the graph bends.

A filament lamp does not obey Ohm’s law — I against V curves away from a straight line
Resistance rises with temperature, and temperature rises with the power dissipated
A gradient taken from a curve only means anything at the point it was taken

Build your own circuit

Place components anywhere and draw the wires yourself. Each wire has a material and a thickness, so the leads have resistance like real ones.

A junction is wherever two ends meet
Wires have resistance too: R = ρL/A
Why connecting leads are thick copper and a heating element is thin nichrome

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)

Current from the supplyI (A)

Terminal p.d. of the supplyV (V)

Resistance of the whole circuitR_{total} (Ω)

Power delivered by the supplyP (W)

Voltage across E1V_{E1} (V)

Current through E1I_{E1} (A)

Power in E1P_{E1} (W)

Voltage across R1V_{R1} (V)

Current through R1I_{R1} (A)

Power in R1P_{R1} (W)

Potential at node 1V_{1} (V)

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

E.m.f. of the cellE (V)

Internal resistance of the cellr (Ω)

Peak voltage of the AC supplyV₀ (V)

Frequency of the AC supplyf (Hz)

Resistance R₁R₁ (Ω)

Resistance R₂R₂ (Ω)

Resistance R₃R₃ (Ω)

Capacitance C₁C₁ (F)

Inductance L₁L₁ (H)

The equations behind it

What dc & ac circuits is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Electric current and circuits

$I = \frac{Δ q}{Δ t}$
Current
$V = I R$
Ohm’s law
$R = \frac{ρ L}{A}$
Resistance of a wire — ρ is the resistivity.
$P = V I = I^{2} R = \frac{V ^{2}}{R}$
Power dissipated
$R_{series} = R_{1} + R_{2} + \dots$
Resistors in series
$\frac{1}{R _{parallel}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$
Resistors in parallel
$ε = I (R + r)$
EMF and internal resistance
$I = n A v q$
Current from drift speed

Capacitance

$C = \frac{q}{V}$
Capacitance
$C = \frac{ε _{0} ε _{r} A}{d}$
Parallel plate capacitor
$E = \frac{1}{2} C V^{2}$
Energy stored
$τ = R C$
Time constant
$q = q_{0} e^{- t / R C}$
Discharging a capacitor