Electricity and d.c. circuits

The emf ε describes the source’s total energy supply per unit charge, while Ir describes the internal part of that transfer. A high-resistance voltmeter draws little current and can therefore estimate the emf. Under a heavier load the current is larger and the voltage drop becomes more noticeable. This model assumes the emf and internal resistance remain approximately constant during the measurement; heating and chemical changes can alter them. If a rechargeable cell is being charged, current direction reverses relative to discharge, so terminal voltage need not be below its emf.

Draw the supply, both components and the output terminals before predicting the response. The divider formula uses the resistance across which the output is taken in the numerator. A connected measuring circuit may load that branch, so its parallel resistance must be included if the loading is significant. The usual school thermistor is NTC, but not every temperature-dependent resistor follows the same trend. Calibration is needed if the circuit is to measure temperature rather than simply switch at a threshold; the thermistor’s resistance–temperature relation is generally nonlinear.

A resistor transfers energy from the circuit but does not consume charge, so the current does not progressively shrink along one series branch. At a junction, different branch currents can occur because different resistances share the same node-to-node potential difference. For a loop equation, choose a direction and apply one consistent sign convention to emf rises and potential drops. If the algebra gives a negative assumed current, the physical current flows opposite to the arrow initially chosen; the negative result is information rather than evidence that the equation must be discarded.

The axes matter: plotting current vertically against voltage reverses the gradient relationship. Use a suitable range of loads without overheating the source, and take repeated readings to assess scatter. A curve may indicate changing source conditions or an imperfect model rather than a simple arithmetic mistake. Extrapolating to zero current estimates the open-circuit voltage, but the fitted line should be supported by a reasonable measurement range. The magnitude of the gradient has units V/A, equivalent to ohms, providing a dimensional check on the internal-resistance calculation.

First combine the output resistor and load using the parallel-resistance rule. Then use that equivalent resistance in the divider ratio. A voltmeter with resistance much larger than the output branch causes little change, which is why high input resistance is desirable in measurement circuits. A low-resistance device can change the voltage substantially and may make the divider unsuitable as a power supply. The same idea explains why connecting a new sensor or meter can alter the quantity being measured; the measuring instrument becomes part of the circuit.

A longer wire of unchanged cross-section offers more resistance, while a larger cross-section provides more paths for charge flow. These geometry changes do not by themselves change the material’s resistivity if its condition remains the same. Stretching a real wire, however, can alter several properties simultaneously, so read the assumptions carefully. In a metal, raising temperature usually increases resistivity as lattice vibrations increase scattering of charge carriers. During a practical resistivity measurement, a large current can heat the wire and undermine a comparison intended to vary only its length.