Current and circuits

📚The physics

Current is the rate of flow of charge, \( I = \Delta q/\Delta t \), measured in amperes. Conventional current is defined as flowing from positive to negative, which is opposite to the actual motion of the electrons — an accident of history that has never been corrected and never causes a problem provided you stay consistent.

emf and potential difference are not the same quantity. Electromotive force is the energy given to each coulomb by the source; potential difference is the energy delivered by each coulomb to a component. The difference between them is the energy wasted inside the source itself, in its internal resistance \(r\).

This is why a car’s headlights dim when the starter motor turns: the huge current through the starter causes a large \(Ir\) drop inside the battery, so less is left for everything else.

Resistance is \( R = V/I \), always. Ohm’s law is the stronger claim that \(R\) stays constant as \(V\) changes, and it only holds for an ohmic conductor at constant temperature. Three \(I\)–\(V\) characteristics are worth recognising instantly: an ohmic conductor gives a straight line through the origin; a filament lamp gives a curve that flattens, because heating raises the resistance; a diode conducts sharply above about 0.6 V in forward bias and essentially not at all in reverse.

Resistivity \(\rho\) is the material property behind resistance: \( R = \rho L/A \). Longer wire means more resistance, thicker wire means less. Note that \(A\) is the cross-sectional area, so doubling the diameter quarters the resistance, not halves it.

Combining resistors. In series the current is the same everywhere and \( R_{\text{total}} = R_1 + R_2 + \ldots \) In parallel the potential difference is the same across each branch and \( 1/R_{\text{total}} = 1/R_1 + 1/R_2 + \ldots \) Two sanity checks that catch most arithmetic errors: a parallel combination is always smaller than the smallest individual resistor, and a series combination is always larger than the largest.

Power comes in three forms of the same statement:

Choose whichever matches what you already know. The \(I^{2}\) version explains why transmission lines run at very high voltage: for a given power, higher voltage means lower current, and losses scale as the square of the current.

Potential dividers split a supply voltage in the ratio of the resistances: \( V_{\text{out}} = V_{\text{in}} \times R_2/(R_1 + R_2) \). Replace one resistor with a thermistor or an LDR and the output voltage becomes a temperature or light sensor, which is the basis of most simple electronic control circuits.