Electricity — Notes

Electric current is the rate of flow of charge: $I = \dfrac{Q}{t}$. SI unit ampere (1 A = 1 C/s); 1 mA $= 10^{-3}$ A, 1 μA $= 10^{-6}$ A. In metal wires the moving charges are electrons; conventional current is taken from the + terminal to the − terminal, opposite to electron flow. Charge of an electron $e = 1.6 \times 10^{-19}$ C; $Q = n e$, and 1 C is the charge of about $6.25 \times 10^{18}$ electrons.

Potential difference between two points is the work done to move a unit charge between them: $V = \dfrac{W}{Q}$. SI unit volt (1 V = 1 J/C). A cell maintains a potential difference by chemical action.

Measuring: an ammeter (very low resistance) is connected in series; a voltmeter (very high resistance) in parallel across the component. The + terminal of each meter is connected towards the + terminal of the cell.

Circuit symbols: cell (long thin line +, short thick line −), battery, plug key (open/closed), wire joint, wires crossing without joining, bulb, resistor (zigzag), rheostat (zigzag with arrow), ammeter, voltmeter.

Ohm’s law: the potential difference across a metallic wire is directly proportional to the current through it, provided its temperature remains the same: $V = IR$.

Resistance $R = \dfrac{V}{I}$ is the opposition a conductor offers to the flow of charge. SI unit ohm (1 Ω = 1 V/1 A). The V–I graph of an ohmic conductor is a straight line through the origin; its slope (V on the y-axis) is $R$, so a steeper line means a larger resistance.

Verification: battery, plug key, ammeter, nichrome wire and rheostat in series, voltmeter across the wire; vary the current with the rheostat, record V and I, plot V against I. Keep currents small so the wire does not heat up.

A rheostat (variable resistance) changes the current without changing the source. Good conductors have low resistance; insulators very high.

At a given temperature, $R \propto l$ and $R \propto \dfrac{1}{A}$, so $R = \rho \dfrac{l}{A}$. Resistivity $\rho$ depends on the material and temperature, not on the length or thickness; SI unit Ω m.

• Cutting a wire into $n$ equal pieces gives $\dfrac{R}{n}$ each.
• Stretching a wire to $n$ times its length (volume constant) makes its area $\dfrac{A}{n}$, so $R$ becomes $n^2 R$.
• Metals and alloys: about $10^{-8}$ to $10^{-6}$ Ω m; insulators: about $10^{12}$ to $10^{17}$ Ω m.
• Alloys have higher resistivity than their constituent metals and do not oxidise readily at high temperature, so they are used in heating devices (nichrome). Copper and aluminium (low resistivity) are used for wires; tungsten (very high melting point) for filaments.

Remember to convert mm² to m² by multiplying by $10^{-6}$.

Series: the same current flows through each resistor; potential differences add: $V = V_1 + V_2 + V_3$. Equivalent resistance $R_s = R_1 + R_2 + R_3$, larger than the largest. A break anywhere stops the whole circuit.

Parallel: the same potential difference acts across each resistor; currents add: $I = I_1 + I_2 + I_3$. $\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$, smaller than the smallest. For two resistors, $R_p = \dfrac{R_1 R_2}{R_1 + R_2}$.

Combined circuits: replace each parallel or series group by its equivalent, find the total current, then work back to each current and potential difference. Check that voltages add up to the supply and currents add up at junctions.

Household wiring is parallel: each appliance gets the full 220 V, draws the current it needs and can be switched independently; in series, one fault would stop everything and the voltage would be shared.

The energy supplied by the source in driving current through a resistor is dissipated as heat: $H = VIt = I^2Rt$ (Joule’s law of heating). Heat is proportional to the square of the current, to the resistance and to the time.

Applications: electric iron, toaster, kettle and room heater (nichrome coils); incandescent bulb — a tungsten filament (melting point about 3380 °C) glows white-hot in a bulb filled with argon or nitrogen to prolong its life; most of the energy still becomes heat. Electric fuse — a wire of low melting point connected in series; it melts when the current exceeds the safe value, breaking the circuit. Common ratings: 1, 2, 3, 5, 10 and 15 A; choose the rating just above the normal current $I = P/V$.

Unwanted heating wastes energy in wires, motors and electronic devices and can cause fires in overloaded circuits.

Electric power is the rate at which electrical energy is used: $P = \dfrac{W}{t} = VI = I^2R = \dfrac{V^2}{R}$. SI unit watt (1 W = 1 J/s = 1 V × 1 A); 1 kW = 1000 W.

Energy $E = Pt$. Commercial unit: kilowatt hour (one "unit") $= 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^6$ J. The kWh is a unit of energy, not power.

Bills: energy per appliance = power × number × hours; add, convert Wh to kWh, multiply by days and by the rate per unit.

Ratings: "P watts, V volts" means P is used only at V. For a fixed voltage, $R = V^2/P$: a lower-wattage bulb has a higher resistance. In parallel, each gets the full voltage and the higher-wattage bulb is brighter; in series, the same current flows and the bulb with the higher resistance (lower wattage) is brighter.