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Current Electricity

14 min read

Kirchhoff's laws

Complex circuits are solved with two laws. Kirchhoff’s first law (junction rule): the total current entering a junction equals the total current leaving it. This is conservation of charge. Kirchhoff’s second law (loop rule): in any closed loop, the sum of the e.m.f.s equals the sum of the potential drops (IR). This is conservation of energy.

Choose a direction for each current and a direction round each loop; a negative answer means the current is the opposite way.
Worked exampleA simple loop

A 12 V battery with negligible resistance is connected to a 4 Ω and a 2 Ω resistor in series. Find the current and the voltage across the 4 Ω resistor.

  1. Total resistance 6 Ω.
  2. Voltage across 4 Ω.

Answer: The current is 2 A and the voltage across the 4 Ω resistor is 8 V.

Internal resistance and potential dividers

A real source has internal resistance r. A potential divider of two resistors gives a fraction of the input voltage: the lower resistor R₂ gives V_out.
Try itA potential divider with a source of internal resistance

Choose the e.m.f. and internal resistance of a cell and the load resistance. Find the current, the terminal voltage and the power lost in the cell.

Open the Ohm's law lab
Measure the current for different voltages.
Ohm’s law graphA graph of current against voltage for a resistor. The line is straight and passes through the origin, so current is proportional to voltage. A steeper line means a smaller resistance.voltage V (volts)current I (A)small Rlarge Rstraight line through 0:I ∝ Vgradient = 1 ÷ R

Resistivity, temperature and measuring instruments

The resistivity of a metal rises with temperature, because the lattice vibrations scatter the electrons more: R = R₀(1 + αΔT). In a thermistor the resistance falls as the temperature rises, used in temperature sensors. A light-dependent resistor (LDR) has a resistance that falls as light increases. A voltmeter has a very high resistance and is connected in parallel; an ammeter has a very low resistance and is connected in series. A galvanometer with a series resistor makes a voltmeter and with a parallel shunt makes an ammeter.

InstrumentUse
Wheatstone bridgeBalanced when R₁/R₂ = R₃/R₄: finds an unknown resistance accurately
PotentiometerCompares e.m.f.s and measures e.m.f. without drawing current
Metre bridgeA practical Wheatstone bridge using a uniform wire

Capacitors in circuits

Charging and discharging a capacitorTwo graphs of voltage against time for a capacitor with a resistor. While discharging, the voltage falls exponentially. While charging it rises towards the supply voltage. In both, the time constant RC is the time for the voltage to change by 63 percent of its total change; after five time constants the change is practically complete.time tVdischarging: V = V₀ e^(−t/RC)charging: V = V₀ (1 − e^(−t/RC))RCV₀
The time constant τ = RC is the time for the charge or voltage to fall to 37% (1/e) of its starting value.
Try itDischarging a capacitor

Choose the resistance, the capacitance and the time. Find the time constant and the voltage left on a capacitor that started at 10 V.

The current is the same all the way round a series circuit.
Check yourself Kirchhoff's first law is a statement of conservation of…
Check yourself A cell of e.m.f. 6 V and internal resistance 1 Ω drives 2 A through a resistor. The terminal voltage is…
Check yourself A 100 kΩ resistor and 10 µF capacitor have a time constant of…

Key terms

Kirchhoff’s first law
The sum of currents into a junction equals the sum out of it.
Internal resistance
The resistance inside a source that lowers its terminal voltage when current flows.
Time constant
RC: the time for a discharging capacitor’s voltage to fall to 1/e of its initial value.
Thermistor
A resistor whose resistance changes with temperature.

Summary

  • Kirchhoff: ΣI in = ΣI out; Σe.m.f. = ΣIR in a loop.
  • Terminal voltage V = E − Ir; potential divider V_out = V_in R₂ ÷ (R₁ + R₂).
  • Metal resistance rises with temperature; thermistors and LDRs are sensors.
  • RC circuits: τ = RC and V = V₀ e^(−t/RC).
Electromagnetism Electronics