4.1 Electric current

Definition of electric current

Electric current describes the flow of electric charge through a conductor. It's the starting point for understanding circuits, and you'll rely on it throughout electromagnetics.

Charge flow in conductors

Current occurs when free charge carriers move through a conductive material. In metals, those carriers are free electrons. In electrolytes, they're ions dissolved in solution.

• An electric potential difference (voltage) across the conductor drives this flow
• Current is measured in amperes (A), where 1 A = 1 coulomb of charge passing a point per second: $I = \frac{\Delta Q}{\Delta t}$
• A helpful analogy: charge is like water, voltage is like pressure, and the conductor is the pipe. More pressure means more flow.

Current direction conventions

• Conventional current flows from the positive terminal to the negative terminal
• Electron flow moves in the opposite direction, from negative to positive
• This mismatch exists because Benjamin Franklin guessed wrong about which charges move before electrons were discovered. The convention stuck.
• Most physics and engineering courses use conventional current unless stated otherwise. Pay attention to which convention your instructor uses.

Microscopic model of current

At the atomic level, electrons in a metal are constantly bouncing around randomly at high thermal speeds. When you apply a voltage, they gain a small net velocity in one direction on top of all that random motion. That net velocity is what produces current.

Drift velocity of electrons

Drift velocity ($v_d$) is the average velocity of charge carriers due to an applied electric field. It's surprisingly slow.

$v_d = \frac{I}{nAq}$

where:

• $I$ = current
• $n$ = number density of charge carriers (carriers per cubic meter)
• $A$ = cross-sectional area of the conductor
• $q$ = charge per carrier ($1.6 \times 10^{-19}$ C for electrons)

In a typical copper wire carrying 1 A, drift velocity is on the order of $10^{-4}$ m/s. So why does a light turn on instantly when you flip a switch? Because the electric field propagates through the wire at nearly the speed of light, pushing all the electrons into motion almost simultaneously.

Current density

Current density ($\vec{J}$) is a vector quantity that describes how much current flows per unit cross-sectional area.

$J = nqv_d$

To get the total current from current density, you integrate over the cross-sectional area:

$I = \int \vec{J} \cdot d\vec{A}$

For a uniform current distribution, this simplifies to $I = JA$. Current density becomes important when the current isn't spread evenly across a conductor.

Ohm's law

Ohm's law is the most-used relationship in circuit analysis:

$V = IR$

It says the voltage across a conductor equals the current through it times its resistance. Rearranging: $I = \frac{V}{R}$.

Resistance vs. conductance

• Resistance ($R$) quantifies how much a material opposes current flow. Measured in ohms ($\Omega$).
• Conductance ($G$) is the inverse: how easily current flows. Measured in siemens (S).

$G = \frac{1}{R}$

So Ohm's law can also be written as $I = GV$. You'll mostly use the $V = IR$ form, but conductance shows up in some circuit analysis methods.

Temperature effects on resistance

Resistance isn't fixed; it changes with temperature.

• Metals: resistance increases with temperature. Higher temperature means more lattice vibrations, which scatter electrons more.
• Semiconductors: resistance decreases with temperature. Heat frees more charge carriers, which outweighs the increased scattering.

The temperature dependence for metals is approximated by:

$R = R_0[1 + \alpha(T - T_0)]$

where $R_0$ is resistance at reference temperature $T_0$ and $\alpha$ is the temperature coefficient of resistance. For copper, $\alpha \approx 3.9 \times 10^{-3} \, \text{°C}^{-1}$.

Superconductors are a special case: below a critical temperature, their resistance drops to exactly zero.

Electrical power

Power in resistive circuits

Power is the rate at which energy is transferred or dissipated in a circuit element. For any component:

$P = VI$

For resistors specifically, you can substitute Ohm's law to get three equivalent forms:

$P = VI = I^2R = \frac{V^2}{R}$

Power is measured in watts (W), where 1 W = 1 J/s. Use whichever form is most convenient based on what quantities you know.

Joule heating

When current flows through a resistor, electrical energy converts to thermal energy. This is Joule heating (also called resistive or ohmic heating), and the heat generated per unit time is:

$P = I^2R$

This effect is useful in electric heaters, toasters, and incandescent bulbs. It's also why fuses work: too much current generates enough heat to melt the fuse wire and break the circuit. In high-power electronics, Joule heating is a problem that requires active cooling.

Direct vs. alternating current

DC sources and applications

Direct current (DC) flows in one direction with constant (or nearly constant) magnitude.

• Sources: batteries, solar cells, DC power supplies
• Applications: low-voltage electronics, LED lighting, electric vehicles
• DC is the natural choice for energy storage in batteries and capacitors

AC characteristics and frequency

Alternating current (AC) periodically reverses direction and oscillates in magnitude, typically following a sine wave:

$I(t) = I_0 \sin(2\pi ft)$

where $I_0$ is the peak current and $f$ is the frequency.

• Power grids use AC at 60 Hz (North America) or 50 Hz (most of the rest of the world)
• AC's major advantage: transformers can easily step voltage up or down, making long-distance power transmission efficient
• Most wall outlets deliver AC; devices that need DC use internal rectifiers to convert it

Current in different circuit elements

Current through resistors

• Follows Ohm's law: $I = \frac{V}{R}$
• Current and voltage are in phase (they rise and fall together for AC)
• All energy dissipated as heat
• Resistors are used for current limiting and voltage division

Current through capacitors

• Current through a capacitor depends on how fast the voltage is changing: $I = C\frac{dV}{dt}$
• For AC, current leads voltage by 90°
• A fully charged capacitor blocks DC entirely (no steady-state DC current flows through it)
• For a sinusoidal voltage, the AC current magnitude is $I = 2\pi fCV$, so higher frequencies pass more easily

Current through inductors

• Voltage across an inductor depends on how fast the current is changing: $V = L\frac{dI}{dt}$
• For AC, current lags voltage by 90°
• Inductors oppose changes in current, not current itself
• For a sinusoidal voltage, the AC current magnitude is $I = \frac{V}{2\pi fL}$, so higher frequencies are resisted more

Kirchhoff's current law

Junction rule

At any junction (node) in a circuit, the total current flowing in equals the total current flowing out:

$\sum I_{in} = \sum I_{out}$

This is a direct consequence of conservation of charge: charge can't pile up at or vanish from a junction. It applies to both DC and AC circuits at every instant in time.

Applications in circuit analysis

• Use KCL to write equations at each junction when solving complex circuits
• Combine with Kirchhoff's voltage law (KVL, the loop rule) to get enough equations to solve for all unknowns
• Particularly useful for analyzing parallel branches and current dividers
• For a circuit with $N$ nodes, KCL gives you $N - 1$ independent equations

Measuring electric current

Ammeters and their use

• An ammeter must be connected in series with the component you're measuring
• An ideal ammeter has zero resistance so it doesn't change the current it's trying to measure. Real ammeters have very small but nonzero resistance.
• Always select the appropriate range before measuring. Starting on the highest range and working down protects the meter from overcurrent.

Safety considerations

• Never connect an ammeter in parallel across a voltage source. Its near-zero resistance would create a short circuit, potentially damaging the meter or the source.
• Use fused test leads to protect against accidental overcurrent
• Ensure proper insulation and grounding when working with high currents
• Know the maximum current rating of both the ammeter and the circuit components

Current density and continuity

Current density vector

The current density vector relates directly to the electric field inside a conductor:

$\vec{J} = \sigma \vec{E}$

where $\sigma$ is the conductivity of the material (in S/m) and $\vec{E}$ is the electric field. This is actually the microscopic form of Ohm's law. Units of $\vec{J}$ are A/m².

Continuity equation

The continuity equation expresses conservation of charge in differential form:

$\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0$

where $\rho$ is the charge density. In plain terms: if more current flows out of a region than flows in, the charge density inside that region must be decreasing. For steady currents (the common case in this course), $\frac{\partial \rho}{\partial t} = 0$, so $\nabla \cdot \vec{J} = 0$.

Superconductivity

Zero resistance phenomenon

Certain materials lose all electrical resistance when cooled below a critical temperature ($T_c$). This was first observed in mercury ($T_c = 4.2$ K) by Heike Kamerlingh Onnes in 1911.

• The mechanism involves electrons forming Cooper pairs that move through the lattice without scattering
• Superconductors also exhibit perfect diamagnetism (the Meissner effect): they expel magnetic fields from their interior entirely

Applications of superconductors

• MRI machines and particle accelerators use superconducting electromagnets to generate extremely strong, stable magnetic fields
• SQUIDs (Superconducting Quantum Interference Devices) detect incredibly small magnetic fields
• Superconducting power lines could transmit electricity with zero resistive losses
• Active research areas include quantum computing and maglev transportation

Electrolytic conduction

Ionic current in solutions

In electrolyte solutions, current is carried by the movement of ions rather than free electrons. Positive ions (cations) drift toward the cathode, and negative ions (anions) drift toward the anode.

• The amount of current depends on ion concentration, ion mobility, and the applied electric field
• This type of conduction is central to how batteries, electroplating systems, and electrochemical sensors work

Faraday's laws of electrolysis

Faraday's laws connect the amount of charge passed through an electrolyte to the mass of substance deposited:

$m = \frac{QM}{zF}$

where:

• $Q$ = total charge passed (in coulombs)
• $M$ = molar mass of the substance
• $z$ = valence number (electrons transferred per ion)
• $F$ = Faraday constant ($\approx 96{,}485$ C/mol)

The first law says mass deposited is proportional to total charge. The second law says that for the same charge, different substances deposit in proportion to their equivalent weights ($M/z$).

Semiconductor current

Electron vs. hole current

In semiconductors, two types of charge carriers contribute to current:

• Electrons in the conduction band move in the direction opposite to the electric field
• Holes in the valence band behave like positive charges moving with the field

Total current is the sum of both contributions: $I_{total} = I_e + I_h$. Electron mobility is generally higher than hole mobility in most semiconductors (for silicon, roughly 3x higher).

Doping effects on conductivity

Doping means intentionally adding impurity atoms to a semiconductor to control its conductivity.

• N-type doping (e.g., adding phosphorus to silicon) adds extra electrons as majority carriers
• P-type doping (e.g., adding boron to silicon) creates extra holes as majority carriers
• Even small amounts of dopant dramatically increase conductivity. Pure silicon has about $10^{10}$ carriers/cm³; doped silicon can have $10^{15}$ to $10^{18}$ carriers/cm³.
• Combining n-type and p-type regions creates p-n junctions, the basis of diodes and transistors

Plasma currents

Ionized gas conduction

A plasma is a partially or fully ionized gas where both free electrons and ions carry current. Unlike metals, plasmas exhibit complex, nonlinear behavior because the charged particles interact with each other and with electromagnetic fields collectively.

Applications in technology

• Fluorescent lamps and plasma displays use controlled plasma conduction
• Plasma cutting and welding exploit the extreme temperatures of plasma arcs
• Fusion energy research (tokamaks, stellarators) confines plasma with magnetic fields to sustain nuclear fusion reactions
• Ion engines and plasma thrusters provide efficient propulsion for spacecraft

Biological electric currents

Nerve impulses

Neurons transmit signals through action potentials: rapid, temporary changes in the voltage across the cell membrane caused by ion channels opening and closing in sequence.

• Typical amplitude: about 100 mV
• Typical duration: 1-2 ms
• These electrical pulses propagate along the axon, enabling communication throughout the nervous system

Electrocardiograms and brain activity

• An ECG (electrocardiogram) measures the electrical activity of the heart over time, detecting the coordinated depolarization and repolarization of cardiac muscle
• An EEG (electroencephalogram) records electrical activity of the brain through electrodes on the scalp
• Both techniques rely on sensitive voltage and current measurements and are essential diagnostic tools in medicine

Current in electromagnetic fields

Displacement current

Maxwell introduced displacement current to fix a gap in Ampère's law. Even when no physical charges are flowing (like between the plates of a charging capacitor), a changing electric field acts as if there's a current:

$I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$

where $\varepsilon_0$ is the permittivity of free space and $\Phi_E$ is the electric flux. This concept was essential for predicting electromagnetic waves.

Maxwell's equations and current

Ampère's law, with Maxwell's correction, states that the curl of the magnetic field depends on both conduction current and displacement current. Including displacement current allowed Maxwell to show that changing electric and magnetic fields can sustain each other and propagate as electromagnetic waves. This unified electricity and magnetism into a single theoretical framework.