Electric Circuits

Current tells you how much charge flows past a point in a wire each second, $I = \frac{dq}{dt}$, and it’s driven by a potential difference called the emf (electromotive force), such as a battery’s. In a metal it comes from electrons drifting slowly along, $I = nqv_dA$, but conventional current points the way positive charge would move. Current density $\vec{J}$ is current per unit area, and when it varies across a wire you integrate it over the area to get the total current.

A circuit is a closed loop that charge can flow around, built from batteries, wires, resistors, bulbs, capacitors, inductors, switches and meters, each drawn with a standard schematic symbol. In an open circuit a break stops the flow, and a short circuit is a path with no change in potential, so current takes it and skips the elements it bypasses.

Resistance tells you how hard it is to push current through an object. For a uniform wire $R = \frac{\rho \ell}{A}$, so longer, thinner wires and higher-resistivity materials resist more, and a conductor’s resistivity usually rises with temperature. Ohm’s law, $\Delta V = IR$, holds for ohmic materials, and the slope of a current-versus-voltage graph for one is $\frac{1}{R}$.

Power is the rate a circuit element transfers energy, $P = I\Delta V$, which for a resistor becomes $P = I^2R = \frac{(\Delta V)^2}{R}$. A bulb that uses more power glows brighter, so comparing power lets you rank the brightness of bulbs in a circuit.

Resistors in series carry the same current and their resistances add, $R_{eq} = R_1 + R_2 + \cdots$. In parallel they share the same potential difference and $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$, so adding branches lowers the total. A real battery acts like an ideal emf in series with an internal resistance r, so its terminal voltage drops when current flows, $\Delta V = \mathcal{E} - Ir$; ammeters go in series and voltmeters in parallel.

Kirchhoff’s loop rule says the potential differences around any closed loop add to zero, $\sum \Delta V = 0$, because a charge that travels all the way around ends with the energy it started with. You use it to write equations for unknown currents, and you can graph the electric potential at each point as you travel around a loop.

Kirchhoff’s junction rule says the current flowing into a junction equals the current flowing out, $\sum I_{in} = \sum I_{out}$, because charge can’t pile up or disappear there. Together with the loop rule it lets you solve a multi-loop circuit as a set of simultaneous equations.

Capacitors in parallel add, $C_{eq} = C_1 + C_2 + \cdots$, while in series $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$ and each one holds the same charge. In an RC circuit the loop rule gives a differential equation whose solution is exponential, with time constant $\tau = RC$: a charging capacitor reaches about 63% of its final charge after one time constant, and a discharging one drops to about 37%. An uncharged capacitor first acts like a wire, and after a long time no current flows in its branch.