By convention, current is drawn in the direction positive charges would move. In many metal wires, the actual moving charges are electrons, so the electron motion is opposite the conventional current direction. The standard unit of current is coulombs per second $(C / s)$ , also called the ampere.

Electromotive force, voltage

Electromotive force (emf) isn’t a literal force. It’s the potential difference a source provides, measured in volts. A battery is a common source of emf.

In an ideal battery with no internal resistance, the potential difference across the terminals equals the emf. Real batteries have internal resistance, so when current flows, part of the emf is used up as a voltage drop inside the battery. That reduces the voltage you measure at the terminals.

The terminal potential (terminal voltage) is the measurable voltage across a battery’s external terminals when it’s connected to a circuit. A useful model is an ideal voltage source in series with a small internal resistance (like a resistor inside the battery). Because current through that internal resistance causes a voltage drop, the terminal potential is

Terminal potential $= EMF - I \cdot R_{in t er na l}$ , where $I$ is the current.

That’s why a battery’s voltage under load is lower than its nominal emf.

Resistance

Ohm’s law relates current $(I)$ , voltage $(V)$ , and resistance $(R)$ :

$I = V / R$ .

Resistors are commonly connected in series or parallel.

For resistors in series:

The same current flows through each resistor.
The total voltage is the sum of the individual voltage drops.
Each drop follows $V = I \cdot R$ (equivalently, $I = V / R$ ).

For resistors in parallel:

The voltage across each resistor is the same.
The total current is the sum of the currents in each branch.
A branch with higher resistance carries lower current.

A material’s resistivity $(ρ)$ connects the physical dimensions of a conductor to its resistance:

$R = ρ L / A$ ,

where $L$ is the conductor length and $A$ is its cross-sectional area. Resistance decreases when resistivity is lower, the conductor is shorter, or the cross-sectional area is larger. That’s why extension cords use thick wires: lower resistance means less heating and less energy loss.

Resistors in series and parallel combinations

Capacitance

A capacitor stores electrical energy by separating charge on two conductive plates with an insulating gap between them. For a parallel-plate capacitor, the capacitance is

$C = Q / V = ϵ A / d$

where:

$Q$ is the stored charge,
$V$ is the voltage across the plates,
$A$ is the plate area,
$d$ is the plate separation,
$ϵ$ is the permittivity of the material between the plates.

The voltage can also be written as $V = E d$ , where $E$ is the electric field between the plates.

The energy stored in a charged capacitor is

$U = \frac{1}{2} Q Δ V = \frac{1}{2} C (Δ V)^{2} = \frac{Q ^{2}}{2 C}$

where $U$ is the potential energy.

Capacitors can be connected:

In series, where the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances.
In parallel, where the equivalent capacitance is the sum of the individual capacitances.

Placing a dielectric (a nonconducting material) between the plates increases capacitance. You can think of this as either:

allowing more charge to be stored for the same voltage, or
reducing the voltage for a fixed amount of charge.

With a dielectric, the voltage becomes $V = V_{0} / κ$ and the capacitance becomes $C = κ C_{0}$ , where $κ$ is the dielectric constant.

Discharge of a capacitor

During discharge, a capacitor releases stored charge by driving a current through an external resistor. In this situation, the capacitor acts like a temporary battery, supplying energy to the circuit as it loses charge. As discharge continues, the current decreases over time because less stored energy remains to push charge through the circuit.

Conductivity

Conductivity describes how easily a material carries electrical current.

In solutions, electrolyte concentration matters:

With no electrolytes, there’s no ionization, so there’s no conductivity.
At an optimal concentration, ions move most freely, so conductivity is highest.
At very high concentration, ions crowd together, mobility drops, and conductivity decreases.

Temperature also affects conductivity:

In metals, higher temperature decreases conductivity because increased lattice vibrations scatter electrons more.
In semiconductors, higher temperature typically increases conductivity by creating more charge carriers.
At extremely low temperatures, some materials become superconductors, meaning current flows with no resistance.

Mathematically, conductivity $(σ)$ is the inverse of resistivity $(ρ)$ . In practice, conductivity can be measured by placing a capacitor in a solution; the current that flows between the plates reflects how well the solution conducts electricity.

Meters

An ammeter measures the current flowing through a circuit.

An ohmmeter measures the resistance of a component.

A probe is the part of a measuring instrument that connects the meter to the circuit or component being tested.

A voltmeter measures the voltage (electric potential difference) between two points in a circuit.

Magnetism

Definition of magnetic field $B$

The magnetic field $(B)$ is a vector field that exists in regions where a moving charge experiences a force due to its motion. When a charged particle moves through a magnetic field, it can be deflected by a force whose magnitude and direction depend on the particle’s velocity and the field strength. Magnetic field strength is measured in teslas $(T)$ , which can also be written as $N \cdot s / m \cdot C$ .

Motion of charged particles in magnetic fields; Lorentz force

When a charged particle moves through a magnetic field, it experiences an electromagnetic force given by

$F = q v B sin θ$

where:

$q$ is the charge,
$v$ is the velocity,
$B$ is the magnetic field,
$θ$ is the angle between the velocity and the magnetic field.

This force is always perpendicular to both the direction of motion and the magnetic field. To find its direction, use the right hand rule: point your thumb in the direction of a positive charge’s velocity, your middle finger in the direction of the magnetic field, and your palm faces the direction of the force.

If the charge moves in a circular path, the electromagnetic force provides the centripetal force needed to keep the particle in orbit, so it satisfies $F = m v^{2} / r$ .

For current-carrying wires, the magnetic force can be written as $F = i L B sin θ$ , where $i$ is the current and $L$ is the length of the wire. Here, current is treated as the motion of positive charges. Using the same right hand rule shows that wires with currents in the same direction attract each other, while wires with currents in opposite directions repel.

Current

Rate of charge flow: $I = Δ Q /Δ t$
Conventional current direction: movement of positive charges
Unit: ampere $(A = C / s)$

Electromotive force, voltage

EMF: potential difference provided by a source (volts)
Terminal potential: $EMF - I \cdot R_{in t er na l}$
Real batteries: voltage under load is less than nominal emf due to internal resistance

Resistance

Ohm’s law: $I = V / R$
Series resistors: same current, total voltage is sum of drops
Parallel resistors: same voltage, total current is sum of branch currents
Resistivity: $R = ρ L / A$
- Lower resistivity, shorter length, larger area $\to$ lower resistance

Capacitance

Capacitor: stores charge on two plates separated by insulator
Capacitance: $C = Q / V = ϵ A / d$
Energy stored: $U = \frac{1}{2} Q Δ V = \frac{1}{2} C (Δ V)^{2} = \frac{Q ^{2}}{2 C}$
Series: $1/ C_{e q} = 1/ C_{1} + 1/ C_{2} + \dots$
Parallel: $C_{e q} = C_{1} + C_{2} + \dots$
Dielectric increases capacitance: $C = κ C_{0}$ , $V = V_{0} / κ$

Discharge of a capacitor

Capacitor releases charge through resistor, acting as temporary battery
Discharge current decreases over time as stored energy drops

Conductivity

Conductivity $(σ)$ : ease of current flow, $σ = 1/ ρ$
Electrolyte solutions: optimal concentration maximizes conductivity
- Too low: no ions; too high: ion crowding reduces mobility
Temperature effects:
- Metals: higher temperature $\to$ lower conductivity
- Semiconductors: higher temperature $\to$ higher conductivity
- Superconductors: zero resistance at very low temperatures

Meters

Ammeter: measures current
Ohmmeter: measures resistance
Voltmeter: measures voltage (potential difference)
Probe: connects meter to circuit/component

Magnetic field $B$

Vector field affecting moving charges
Measured in teslas $(T)$

Motion of charged particles in magnetic fields; Lorentz force

Force on charge: $F = q v B sin θ$
- Perpendicular to velocity and field (right hand rule)
Circular motion: $F_{ma g} = m v^{2} / r$
Force on current-carrying wire: $F = i L B sin θ$
- Same direction currents attract, opposite repel

Circuit elements, conductivity, and magnetism