Electric circuits

Revision notes

Current is rate of charge flow, I = \(\dfrac{\Delta Q}{\Delta t}\). Electromotive force (emf) is energy supplied per unit charge by a source; potential difference is energy transferred per unit charge between two points. A real source with internal resistance r and emf ε has terminal p.d. V = ε − Ir when delivering current.

Kirchhoff's first law expresses charge conservation at a junction; the sum of currents entering equals the sum leaving. The second law expresses energy conservation around a closed loop. For a uniform conductor R = \(\dfrac{\rho L}{A}\), where ρ is resistivity. Potential dividers use series components to obtain a fraction of a supply voltage, with the output depending on resistance ratios.

Current, charge and potential difference

Current is the rate of flow of charge, I = \(\dfrac{\Delta Q}{\Delta t}\). Potential difference is energy transferred per unit charge, V = \(\dfrac{W}{Q}\). Electromotive force describes energy supplied per unit charge by a source. These definitions are more useful than treating voltage as an unexplained 'push'.

Kirchhoff's first law follows from charge conservation: total current into a junction equals total current out. Kirchhoff's second law follows from energy conservation around a complete loop: the algebraic sum of emfs and potential differences is zero.

Resistance and resistivity

For an ohmic conductor at constant physical conditions, V = IR. Resistance depends on geometry and material. Resistivity is a material property related by R = \(\dfrac{\rho L}{A}\) for a uniform conductor. Increasing length increases resistance; increasing cross-sectional area decreases it.

Temperature can change resistivity. In metals, increased lattice vibration usually increases resistance. A filament lamp therefore has a curved I-V characteristic because its temperature rises as current increases.

Series, parallel and potential dividers

Series resistances add. Parallel combinations satisfy \(\dfrac{1}{R_{\text{total}}}\) = Σ(\(\dfrac{1}{R}\)). In parallel, branch potential differences are equal; currents divide according to branch resistance.

A potential divider produces an output voltage from a fraction of a supply. For two series resistors, \(\dfrac{V_{\text{out}}}{V_{\text{supply}}}\) depends on the resistance across which the output is taken. Sensors such as LDRs or thermistors can make the output depend on environmental conditions.

Internal resistance

A real source has internal resistance r. When current flows, some energy is transferred inside the source and terminal p.d. is lower than emf: V = ε - Ir. A graph of terminal p.d. against current has intercept ε and gradient -r.

Maximum power transfer and source efficiency can be reasoned from how current changes the internal voltage drop; always distinguish power delivered to the load from power dissipated internally.

Electrical power

P = IV. Combining with V = IR gives P = I²R and P = \(\dfrac{V^{2}}{R}\) where appropriate. These forms are not interchangeable without considering what quantity is held constant. Increasing R at fixed current increases power, but increasing R at fixed supply voltage decreases power.

Practical interpretation

When measuring an I-V characteristic, control temperature if the aim is to test ohmic behaviour. For resistivity, measure length and diameter carefully; because area depends on diameter squared, diameter uncertainty can dominate.

Exam technique

Use Kirchhoff's laws with a clear sign convention. Distinguish emf from terminal p.d., resistance from resistivity and current from charge. Before choosing a power equation, identify which circuit quantity is fixed by the situation.

Charge carriers and components

Current in a conductor is I = nAvq, where n is the number of charge carriers per unit volume and v is their drift velocity. Metals have very large n, so drift velocities are small (fractions of a millimetre per second).

An NTC thermistor's resistance falls as its temperature rises, and an LDR's resistance falls as light intensity increases. A semiconductor diode conducts in the forward direction above about 0.6 V and has a very high resistance in reverse.

A superconductor has zero resistivity below its critical temperature.

Potential dividers

For two resistors in series across \(V_{\text{in}}\), the output across R₂ is \(V_{\text{out}}\) = \(\dfrac{V_{\text{in}} \times R_{2}}{R_{1} + R_{2}}\). With a thermistor or LDR as one resistor, the output changes with temperature or light, making a sensor circuit.