3 Current and Circuits

Charge flow and current

Electric current describes how quickly charge flows through a conductor. If charge $\Delta Q$ passes a cross-section during a time interval $\Delta t$, then

The ampere is the unit of current: $1\ \mathrm{A}=1\ \mathrm{C/s}$. In circuit diagrams, conventional current is shown in the direction positive charge would travel, from a source’s positive terminal through the external circuit toward its negative terminal. In metal wires, electrons drift in the opposite direction.

A direct current (DC) flows in one direction. In a steady-state DC circuit, its magnitude is constant.

Takeaway: Current is charge flow per unit time, and conventional current direction is defined by positive-charge motion.

Resistance, voltage, and power

Resistance measures how strongly a component opposes current. For an ohmic resistor, Ohm’s law relates voltage, current, and resistance:

Voltage is measured in volts, current in amperes, and resistance in ohms, with $1\ \Omega=1\ \mathrm{V/A}$. For a uniform wire, resistance depends on its resistivity, length, and cross-sectional area:

A longer wire has greater resistance, while a thicker wire has less. The electrical power converted by a resistor can be calculated in equivalent forms:

For example, a $6\ \Omega$ resistor connected across $12\ \mathrm{V}$ carries current $I=\frac{12}{6}=2\ \mathrm{A}$. Its power dissipation is $P=VI=24\ \mathrm{W}$.

Takeaway: Use Ohm’s law to relate voltage, current, and resistance; use a power equation to determine the rate of energy conversion.

Sources and terminal voltage

A battery or generator supplies energy to move charge around a circuit. Its electromotive force (emf), represented by $\varepsilon$, is the energy supplied per unit charge. Despite its name, emf is not a force.

A real source can be modeled as an ideal emf in series with internal resistance $r$. When it supplies current $I$, the terminal voltage is

Thus, while current flows, the voltage across an external load is lower than the source’s emf. With a single load resistor $R$, the total resistance is $R+r$, so the current is

Takeaway: Internal resistance causes a voltage drop inside a real source when it supplies current.

Series and parallel circuits

Circuit structure determines how current and voltage are shared. In a Series circuit, components lie along one path, so the current is the same through each component. Their equivalent resistance is

In a Parallel circuit, components connect across the same two nodes, so the voltage across each branch is the same. The branch currents add, and the equivalent resistance satisfies

For a network built from series and parallel combinations, reduce groups to an equivalent resistance, find the total current using Ohm’s law, and then work back to the individual voltages and currents.

Takeaway: Series components share current; parallel branches share voltage.

Kirchhoff’s rules and a worked example

Kirchhoff’s rules extend circuit analysis to networks that cannot be reduced using series and parallel combinations alone:

1. Junction rule: The total current entering a junction equals the total current leaving it. This expresses conservation of charge.

2. Loop rule: The algebraic sum of voltage changes around any closed loop is zero. This expresses conservation of energy.

To apply the rules, assign current directions, write equations at junctions, and trace independent loops. The chosen current directions may be arbitrary. Across a resistor, moving with the assumed current gives a voltage change of $-IR$; moving against it gives $+IR$. Across a source, moving from its negative terminal to its positive terminal gives $+\varepsilon$, while moving in the reverse direction gives $-\varepsilon$. A negative solved current means the actual direction is opposite the assumed direction.

Example: A $12\ \mathrm{V}$ source with internal resistance $1\ \Omega$ supplies two $4\ \Omega$ resistors in parallel. Let $I$ be the current delivered by the source and $I_1$ and $I_2$ be the branch currents. The junction rule gives

The two equal parallel resistors have equivalent resistance $2\ \Omega$. Including the internal resistance, the total resistance is $3\ \Omega$, so

The terminal voltage is $12-(4)(1)=8\ \mathrm{V}$. Each branch therefore carries $8/4=2\ \mathrm{A}$, and the branch currents sum to the source current.

Takeaway: Use the junction rule to track current and the loop rule to track voltage changes; both follow from conservation principles.