Current Electricity — Notes

• Current is the rate of flow of charge: $I = Q/t$; unit ampere (1 A = 1 C s⁻¹). Charge of an electron $e = 1.6 \times 10^{-19}$ C; number of electrons $n = Q/e$.
• Conventional current flows from + to − in the outside circuit; electrons drift from − to +.
• Potential difference between two points = work done per unit charge moved between them: $V = W/Q$; 1 V = 1 J C⁻¹.
• Resistance = opposition to current: $R = V/I$; 1 Ω = 1 V A⁻¹.
• Ammeter: low resistance, joined in series. Voltmeter: high resistance, joined in parallel.

Ohm’s law: the current through a conductor is directly proportional to the p.d. across its ends, provided temperature and other physical conditions remain unchanged: $V = IR$.

Verification: cell, key, rheostat, ammeter and the resistor in series; voltmeter across the resistor. Vary the current with the rheostat; for each setting record V and I. V ÷ I is constant, and the V–I graph is a straight line through the origin with slope = R.

Ohmic conductors: metals and alloys at constant temperature. Non-ohmic: filament lamp (graph curves as R rises with temperature), junction diode, LED, thermistor, electrolytes.

• $R \propto l$ (longer, more resistance); $R \propto 1/A$ (thicker, less resistance); depends on material and temperature.
• $R = \rho \dfrac{l}{A}$; specific resistance (resistivity) $\rho = \dfrac{RA}{l}$, the resistance of a conductor of unit length and unit area of cross-section; unit Ω m.
• $\rho$ depends on the material and temperature only, not on length or area.
• Area of a round wire $A = \pi d^{2}/4$: halving the diameter makes R four times.
• Stretching a wire to n times its length (volume constant) makes its area 1/n times, so R becomes $n^{2}$ times.
• Copper and aluminium (low ρ): connecting wires and cables. Nichrome (high ρ, high melting point): heating elements. Manganin, constantan (resistance hardly changes with temperature): standard resistors.

• Metals: resistance increases with rise in temperature (the ions vibrate more and impede the electrons).
• Alloys (manganin, constantan): resistance changes very little with temperature.
• Semiconductors (silicon, germanium) and thermistors: resistance decreases as temperature rises.
• Superconductors: materials whose resistance becomes zero below a critical temperature; first seen in mercury below about 4.2 K. A current in a superconducting loop produces no heat, so it can flow without loss.

• E.m.f. (E): work done by the cell per unit positive charge in carrying it round the complete circuit, including the cell; equals the terminal p.d. on open circuit.
• Terminal voltage (V): p.d. across the terminals while the cell supplies current; $V = E - Ir$, so $V < E$.
• Internal resistance (r): resistance of the electrolyte (and electrodes) inside the cell. It increases with the separation of the electrodes and the concentration of the electrolyte; it decreases with a larger electrode area in the electrolyte and a higher temperature.
• $E = I(R + r)$; lost volts = $Ir$.

Series: same current through each; $V = V_{1} + V_{2} + V_{3}$, so $R_{s} = R_{1} + R_{2} + R_{3}$; $R_{s}$ is greater than the largest resistor; one break stops the whole circuit.

Parallel: same p.d. across each; $I = I_{1} + I_{2} + I_{3}$, so $\dfrac{1}{R_{p}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \dfrac{1}{R_{3}}$; $R_{p}$ is less than the smallest; branches work independently. For n equal resistors R: series $nR$, parallel $R/n$.

Networks: reduce each parallel group, add series parts and r, find the main current, then work back with V = IR.