Revision summary

Identify the nodes, the component and the stated conditions. Then choose a relationship that uses the correct current, voltage and source model.

Connections and measurements

• Series components share current along an unbranched path; parallel components share both endpoints and therefore p.d.
• An ammeter measures current in its own path. A voltmeter measures p.d. between its own terminals. Pair component V with component I before using R = V/I.
• Check range, resolution, zero, polarity, loading and thermal conditions. Repeated readings assess scatter but do not remove a consistent wrong-endpoint or zero error.

Resistance and material behaviour

Convert diameter before squaring it. At fixed material, temperature and length, resistance is proportional to 1/A; doubling diameter quarters resistance. On an I-against-V plot, only a straight line through the origin has gradient 1/R at all points.

• Ohmic resistor: proportional I and V under fixed physical conditions.
• Filament lamp: settled heating raises resistance, so the I-V curve becomes less steep at greater voltage magnitude.
• Semiconductor diode: forward conduction and small reverse current before breakdown; no universal forward threshold.
• NTC thermistor: warming lowers resistance. A self-heated voltage sweep differs from a fixed-temperature, low-field measurement.

For a typical metal at the same electric field, greater scattering lowers mean drift speed while carrier density changes little. For the stated NTC regime, increased carrier number density is the main explanation for falling resistivity. Random thermal motion is not the directed drift.

Sources, networks and outputs

These source signs describe current delivered to the load. A terminal-V-against-I line has intercept E_s and gradient -r. Maximum load power occurs at R_external = r for this fixed source model, where load and internal powers are equal; this is not maximum efficiency.

Reduce a network to find total current, then work back to its branch voltages and currents. After a connection changes, recalculate the complete network: another branch's voltage may change even if its resistance does not.

Use the actual supply-terminal p.d. The selected output determines the numerator. Work through sensor resistance, total current and output rather than memorising a direction. An output load is parallel with its selected arm and changes the effective divider.

Capacitors and switching

Parallel capacitors share voltage. In the stated initially uncharged series arrangement, an isolated neutral middle node enforces equal charge magnitudes; its potential need not be zero. Keep any supplied initial node charge in a more general arrangement.

For the simple RC path, τ = RC. Take Q on the initially positive plate, V = Q/C and positive current towards that plate:

At one time constant, a rising quantity reaches 63.2% of its final value and a decaying quantity retains 36.8% of its initial value. The half-value time is τ ln 2, while 90% charging takes -τ ln(0.10).

Read discharge data

Log ratios and exponential arguments are dimensionless. Fit ln(V/V₀) against numeric time; the gradient has unit s^-1. A coarse trapezium sum approximates a curved area, and rounded model values can move a fitted result slightly. Neither is evidence of measured scatter. Meter loading and the actual switched resistor path matter when interpreting a real record.

Quantity and unit reference

Read symbols in context: A as area differs from A as the ampere unit, and C as charge's unit differs from a node labelled C or the capacitance symbol C. Useful prefixes are micro (µ, 10^-6), milli (m, 10^-3), kilo (k, 10³) and mega (M, 10⁶). Squared lengths and areas require squared conversion factors.

For a calculation, draw or trace the nodes, choose reference directions and endpoints, state the component conditions, solve with units, then check current and energy accounts. For a measurement, retain the actual readings and instrument information as well as the inferred result.