Die Druckversion wird nicht mehr unterstützt und kann Darstellungsfehler aufweisen. Bitte aktualisiere deine Browser-Lesezeichen und verwende stattdessen die Standard-Druckfunktion des Browsers.

Electricity and Magnetism

Electricity and Magnetism is a university-level introduction to classical electromagnetism: the study of electric charge, electric and magnetic fields, forces, potentials, induction, circuits, and electromagnetic waves. You will move from experimentally grounded laws such as Coulomb's law and Faraday's law to the compact field description provided by Maxwell's equations. The course emphasizes mathematical reasoning, physical interpretation, model building, and connections to engineering and modern technology.

This aiMOOC is suitable for first- or second-year university students in physics, electrical engineering, applied mathematics, and related programs. You should be comfortable with algebra, trigonometry, basic calculus, vectors, and elementary mechanics. Familiarity with partial derivatives and line, surface, and volume integrals is helpful for the later sections.

By the end of the course, you should be able to:

Electric field: Calculate and interpret fields produced by discrete and continuous charge distributions.
Electric potential: Relate potential, electric field, work, and electrostatic energy.
Magnetic field: Analyze magnetic forces and fields created by currents.
Electromagnetic induction: Predict induced emf using magnetic flux and Faraday's law.
Maxwell's equations: Use integral and differential forms to connect sources, fields, and waves.
Electromagnetic wave: Explain how time-varying electric and magnetic fields propagate energy through space.

Introduction

Electricity and magnetism were once studied as largely separate phenomena. Experiments in the eighteenth and nineteenth centuries revealed a deeper unity: charges create electric fields, moving charges create magnetic fields, changing magnetic fields create electric fields, and changing electric fields contribute to magnetic fields. James Clerk Maxwell assembled these relationships into a coherent theory that predicts electromagnetic waves. Light is one member of this much broader family.

Electromagnetism is central to both fundamental physics and technology. Electric motors, generators, transformers, radio antennas, magnetic resonance systems, power grids, sensors, capacitive touchscreens, particle accelerators, and optical communication all rely on electromagnetic principles. The same mathematical language also develops transferable skills in vector calculus, symmetry arguments, energy methods, and differential equations.

When you solve an electromagnetism problem, ask four questions: What are the sources? What symmetries are present? Which field law matches the geometry? What quantity is actually being requested? This habit helps you choose between direct integration, Gauss's law, Ampère's law, potential methods, or Maxwell's equations.

Mathematical Language and Conventions

This course uses SI units. Charge is measured in coulombs C, electric field in newtons per coulomb N/C or volts per metre V/m, potential in volts V, magnetic field in tesla T, current in amperes A, capacitance in farads F, inductance in henries H, and resistance in ohms.

Vectors are written with bold symbols in prose and vector notation in equations. The gradient $\nabla V$ , divergence $\nabla \cdot 𝐅$ , and curl $\nabla \times 𝐅$ describe how scalar and vector fields vary in space. Surface and line integrals connect local differential laws with global integral laws through the divergence theorem and Stokes' theorem.

Important vacuum constants are the electric constant $ε_{0}$ and magnetic constant $μ_{0}$ . In vacuum, the speed of electromagnetic waves satisfies $c = \frac{1}{\sqrt{μ_{0} ε_{0}}}$ .

Electric Charge and Coulomb's Law

Electric charge is a conserved property of matter. At the microscopic level, electrons carry negative elementary charge and protons carry positive elementary charge of equal magnitude. Charge is quantized in units of the elementary charge $e$ , although macroscopic charge distributions are often treated as continuous.

For two point charges $q_{1}$ and $q_{2}$ separated by displacement vector $𝐫$ , Coulomb's law gives the force $𝐅_{12} = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{r^{2}} \hat{𝐫}$ . The force acts along the line connecting the charges. Like charges repel and unlike charges attract.

The superposition principle is essential: the total force or field is the vector sum of contributions from all sources. For many charges, $𝐄 (𝐫) = \frac{1}{4 π ε_{0}} \sum_{i} q_{i} \frac{𝐫 - 𝐫_{i}}{| 𝐫 - 𝐫_{i} |^{3}}$ .

The Electric Field

The electric field at a point is defined as force per unit positive test charge: $𝐄 = \frac{𝐅}{q}$ . For a point charge $Q$ , $𝐄 (𝐫) = \frac{1}{4 π ε_{0}} \frac{Q}{r^{2}} \hat{𝐫}$ .

Field lines are a visualization tool rather than physical threads. Their tangent gives the local field direction, their density can represent field magnitude, and electrostatic field lines begin on positive charge and end on negative charge or extend to infinity.

For a continuous charge distribution, replace the discrete sum by an integral. Typical source elements are $d q = λ d l$ for a line charge, $d q = σ d A$ for a surface charge, and $d q = ρ d V$ for a volume charge. The resulting field is $𝐄 (𝐫) = \frac{1}{4 π ε_{0}} \int \frac{𝐫 - 𝐫^{'}}{| 𝐫 - 𝐫^{'} |^{3}} d q$ .

Electric Dipoles

An electric dipole consists of equal and opposite charges separated by a small displacement. Its dipole moment points from negative to positive charge: $𝐩 = q 𝐝$ .

Far from an ideal dipole, the electric field decreases approximately as $1 / r^{3}$ , faster than the $1 / r^{2}$ field of a net point charge. A dipole in a uniform external electric field experiences torque $𝝉 = 𝐩 \times 𝐄$ and has potential energy $U = - 𝐩 \cdot 𝐄$ .

Dipoles are important in molecular physics, dielectric materials, antennas, and radiation. The dipole model also illustrates how symmetry and approximation simplify complicated charge distributions.

Electric Flux and Gauss's Law

Electric flux measures how much electric field passes through an oriented surface. For a surface $S$ , $Φ_{E} = \int_{S} 𝐄 \cdot d 𝐀$ . For a closed surface, use the notation $\oint$ .

Gauss's law states $\oint_{S} 𝐄 \cdot d 𝐀 = \frac{Q_{e n c}}{ε_{0}}$ . In differential form, $\nabla \cdot 𝐄 = \frac{ρ}{ε_{0}}$ .

Gauss's law is always true in classical electromagnetism, but it is especially useful for calculating fields when the charge distribution has high symmetry. Spherical symmetry suggests a spherical Gaussian surface, cylindrical symmetry suggests a cylinder, and planar symmetry suggests a pillbox. If the magnitude or direction of $𝐄$ is not sufficiently constrained by symmetry, direct integration or numerical methods may be better.

For a point charge, a spherical Gaussian surface gives $E (4 π r^{2}) = Q / ε_{0}$ , reproducing Coulomb's inverse-square law. For an infinite line charge with linear density $λ$ , a cylindrical Gaussian surface leads to $E = \frac{λ}{2 π ε_{0} r}$ .

Electric Potential and Electrostatic Energy

Electrostatic forces are conservative. The electric potential difference between points A and B is $V_{B} - V_{A} = - \int_{A}^{B} 𝐄 \cdot d 𝐥$ . The electric field is related to potential by $𝐄 = - \nabla V$ .

For a point charge, $V (r) = \frac{1}{4 π ε_{0}} \frac{Q}{r}$ when zero potential is chosen at infinity. Unlike electric field, potential is a scalar, so contributions from several charges add algebraically.

A charge $q$ placed at potential $V$ has electric potential energy $U = q V$ . For two point charges, $U = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{r}$ .

Equipotential surfaces are everywhere perpendicular to the electrostatic field. Moving a charge along an equipotential surface requires no electrostatic work. This connection between geometry and energy is often more efficient than direct force calculations.

Conductors in Electrostatic Equilibrium

In an ideal conductor at electrostatic equilibrium, the electric field inside the conducting material is zero. Any excess charge resides on the surface, and the conductor is an equipotential. Just outside a conductor, the normal component of the electric field is related to surface charge density by $E_{⊥} = σ / ε_{0}$ when the field inside is zero.

Sharp curvature can produce large surface charge density and large local electric fields. This fact is relevant to corona discharge, lightning rods, field emission, and high-voltage engineering.

Capacitance and Dielectrics

A capacitor stores separated charge and electric-field energy. Capacitance is defined by $C = \frac{Q}{Δ V}$ . For an ideal parallel-plate capacitor with plate area $A$ , separation $d$ , and vacuum between the plates, $C = \frac{ε_{0} A}{d}$ when fringing fields are neglected.

Energy stored in a capacitor can be written in equivalent forms: $U = \frac{1}{2} C (Δ V)^{2} = \frac{Q^{2}}{2 C} = \frac{1}{2} Q Δ V$ . In vacuum, the electric-field energy density is $u_{E} = \frac{1}{2} ε_{0} E^{2}$ .

A dielectric is an insulating material that polarizes in an electric field. For a linear dielectric with relative permittivity $ε_{r}$ , the capacitance of a fully filled parallel-plate capacitor becomes $C = ε_{r} ε_{0} A / d$ . The microscopic explanation involves bound charges and induced or permanent molecular dipoles.

Capacitors appear in filters, oscillators, timing circuits, pulsed-power systems, energy storage, sensors, microphones, touch interfaces, and memory devices.

Current, Resistance, and Direct-Current Circuits

Electric current is the rate of charge flow: $I = \frac{d Q}{d t}$ . In a conductor, the current density $𝐉$ describes current per unit area. For a surface, $I = \int 𝐉 \cdot d 𝐀$ .

Microscopically, charge carriers acquire a small drift velocity in response to an electric field. In an ohmic material, $𝐉 = σ 𝐄$ , where $σ$ is conductivity. For a uniform wire of length $L$ , cross-sectional area $A$ , and resistivity $ρ$ , $R = ρ L / A$ . Ohm's law for a circuit element is $V = I R$ when resistance is constant.

Power dissipated by a resistor is $P = I V = I^{2} R = V^{2} / R$ .

Kirchhoff's junction rule expresses charge conservation, while Kirchhoff's loop rule is related to energy conservation in lumped circuits. These rules are highly effective when electromagnetic propagation delays are negligible compared with circuit timescales.

RC Circuits and Time Dependence

In a series RC circuit connected to a constant-voltage source $V_{0}$ , the capacitor charges according to $Q (t) = C V_{0} (1 - e^{- t / R C})$ . The time constant is $τ = R C$ . During discharge, $Q (t) = Q_{0} e^{- t / R C}$ .

The exponential form is a general signature of a first-order linear dynamical system. RC circuits therefore provide a bridge between electromagnetism, differential equations, signal processing, and control systems.

Magnetic Fields and the Lorentz Force

A charge moving with velocity $𝐯$ in electric and magnetic fields experiences the Lorentz force $𝐅 = q (𝐄 + 𝐯 \times 𝐁)$ .

The magnetic part $q 𝐯 \times 𝐁$ is perpendicular to the instantaneous velocity, so a purely magnetic field does no work on an isolated point charge. It can change the direction of motion without changing kinetic energy. For $𝐯 ⊥ 𝐁$ , uniform circular motion has radius $r = \frac{m v}{| q | B}$ . If the velocity also has a component parallel to $𝐁$ , the trajectory is helical.

For a straight wire segment carrying current $I$ in a uniform magnetic field, $𝐅 = I 𝐋 \times 𝐁$ . For a general wire, $d 𝐅 = I d 𝐥 \times 𝐁$ .

Magnetic forces are central to electric motors, mass spectrometers, particle detectors, plasma confinement, loudspeakers, and accelerator beam steering.

Magnetic Fields Produced by Currents

A steady current produces a magnetic field. The Biot-Savart law is $d 𝐁 = \frac{μ_{0}}{4 π} \frac{I d 𝐥 \times \hat{𝐫}}{r^{2}}$ . It plays a role in magnetostatics analogous to Coulomb's law in electrostatics.

For an infinitely long straight wire, $B = \frac{μ_{0} I}{2 π r}$ . The field direction follows the right-hand rule.

Ampère's circuital law for steady currents is $\oint 𝐁 \cdot d 𝐥 = μ_{0} I_{e n c}$ . Like Gauss's law, Ampère's law is most useful for calculation when symmetry makes the field behavior simple.

Solenoids and Magnetic Dipoles

A long ideal solenoid with turn density $n$ and current $I$ has an approximately uniform interior field $B \approx μ_{0} n I$ and a much weaker exterior field.

A current loop has magnetic dipole moment $𝝁 = I A \hat{𝐧}$ . In a uniform magnetic field, the loop experiences torque $𝝉 = 𝝁 \times 𝐁$ and has potential energy $U = - 𝝁 \cdot 𝐁$ .

The dipole picture connects current loops with bar magnets and microscopic magnetic moments. In matter, magnetism arises from quantum-mechanical electron orbital motion and spin, but classical dipole models remain useful for many macroscopic problems.

Magnetic Flux and Electromagnetic Induction

Magnetic flux through a surface is $Φ_{B} = \int_{S} 𝐁 \cdot d 𝐀$ . Faraday's law states that a changing magnetic flux through a circuit induces an electromotive force: $ℰ = - \frac{d Φ_{B}}{d t}$ . For $N$ identical turns, $ℰ = - N \frac{d Φ_{B}}{d t}$ .

The minus sign expresses Lenz's law: the induced effect opposes the change in magnetic flux that produces it. This is a manifestation of energy conservation, not a separate force law.

Flux can change because the magnetic field magnitude changes, the loop area changes, the loop rotates, or the loop moves through a nonuniform field. Motional emf in a conductor moving through a magnetic field can also be analyzed with the magnetic Lorentz force on charge carriers.

Generators convert mechanical work into electrical energy through induction. Transformers rely on time-varying magnetic flux to transfer energy between circuits. Induction also explains eddy-current braking, wireless power transfer, induction heating, and many sensing technologies.

Inductance and RL Circuits

Self-inductance measures how a circuit's own current creates magnetic flux that opposes changes in that current. For a linear inductor, $N Φ_{B} = L I$ and the induced emf is $ℰ_{L} = - L \frac{d I}{d t}$ .

The magnetic energy stored in an inductor is $U_{B} = \frac{1}{2} L I^{2}$ . In vacuum, the magnetic-field energy density is $u_{B} = \frac{B^{2}}{2 μ_{0}}$ .

In a series RL circuit driven by a DC source, the current rises as $I (t) = \frac{V_{0}}{R} (1 - e^{- t R / L})$ , with time constant $τ = L / R$ . Inductors resist rapid changes in current, just as capacitors resist rapid changes in voltage.

Maxwell's Equations

Maxwell's equations summarize classical electromagnetism. In vacuum with charge density $ρ$ and current density $𝐉$ , their differential forms are:

$\nabla \cdot 𝐄 = \frac{ρ}{ε_{0}}$

$\nabla \cdot 𝐁 = 0$

$\nabla \times 𝐄 = - \frac{\partial 𝐁}{\partial t}$

$\nabla \times 𝐁 = μ_{0} 𝐉 + μ_{0} ε_{0} \frac{\partial 𝐄}{\partial t}$

The first equation is Gauss's law for electricity: electric charge is a source or sink of electric field. The second expresses the absence of observed magnetic monopoles in classical electromagnetism: magnetic field lines form closed loops or extend without beginning or end. The third is Faraday's law: changing magnetic fields generate circulating electric fields. The fourth is the Ampère-Maxwell law: currents and changing electric fields generate circulating magnetic fields.

Maxwell's addition of the displacement-current term $μ_{0} ε_{0} \partial 𝐄 / \partial t$ makes the equations consistent with charge conservation and allows electromagnetic waves to propagate through vacuum.

Integral Forms and Boundary Thinking

The integral forms connect fields to finite regions:

$\oint_{S} 𝐄 \cdot d 𝐀 = \frac{Q_{e n c}}{ε_{0}}$

$\oint_{S} 𝐁 \cdot d 𝐀 = 0$

$\oint_{C} 𝐄 \cdot d 𝐥 = - \frac{d}{d t} \int_{S} 𝐁 \cdot d 𝐀$

$\oint_{C} 𝐁 \cdot d 𝐥 = μ_{0} I_{e n c} + μ_{0} ε_{0} \frac{d}{d t} \int_{S} 𝐄 \cdot d 𝐀$

These forms are especially useful when symmetry or boundary geometry matters. Differential forms describe local field behavior at a point, while integral forms describe net behavior over a region.

A powerful strategy is to switch forms deliberately. If a problem emphasizes local source density, a differential equation may be natural. If it emphasizes total enclosed charge, linked current, or flux through a surface, an integral form may reveal the answer more directly.

Electromagnetic Waves

In a source-free region of vacuum, Maxwell's equations imply wave equations such as $\nabla^{2} 𝐄 = μ_{0} ε_{0} \frac{\partial^{2} 𝐄}{\partial t^{2}}$ and $\nabla^{2} 𝐁 = μ_{0} ε_{0} \frac{\partial^{2} 𝐁}{\partial t^{2}}$ .

The propagation speed is $c = 1 / \sqrt{μ_{0} ε_{0}}$ . This result identified light as an electromagnetic wave.

For an ideal plane wave in vacuum, $𝐄$ , $𝐁$ , and the direction of propagation are mutually perpendicular. Their magnitudes satisfy $E = c B$ .

The electromagnetic spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. These categories differ primarily in frequency and wavelength; all are electromagnetic radiation.

Energy transport is described by the Poynting vector $𝐒 = \frac{1}{μ_{0}} 𝐄 \times 𝐁$ . Its direction gives the local direction of electromagnetic energy flow, and its magnitude represents power per unit area for a plane wave.

From Fields to Communication Systems

A time-varying current in an antenna can radiate electromagnetic waves. A receiving antenna converts part of the incident field into electrical signals. Modulation, resonance, bandwidth, impedance matching, and propagation environments determine how efficiently information is transmitted.

The same field theory governs radio communication, Wi-Fi, radar, satellite links, optical fibers, and remote sensing. Engineering models often simplify full Maxwell equations into circuit, transmission-line, or waveguide descriptions depending on the scale and frequency.

Energy, Momentum, and Conservation Laws

Electromagnetic fields store and transport energy. In vacuum, total field energy density is $u = \frac{1}{2} ε_{0} E^{2} + \frac{B^{2}}{2 μ_{0}}$ . The Poynting theorem expresses energy conservation by linking the decrease of field energy inside a volume, the outward flow of electromagnetic energy, and the work done on charges.

Electromagnetic fields also carry momentum. Radiation can exert pressure, a fact observed in optical trapping and important in astrophysics and spacecraft concepts. Classical electromagnetism therefore extends the familiar mechanics ideas of energy and momentum into the field description.

Problem-Solving Strategies

University electromagnetism becomes easier when you separate the physical model from the algebra.

Symmetry in physics: Identify rotational, spherical, cylindrical, planar, or translational symmetry before choosing an equation.
Dimensional analysis: Check that your result has the correct SI units and plausible scaling.
Limiting case: Test what happens when distance becomes very large or small, a parameter goes to zero, or symmetry becomes exact.
Vector calculus: Draw directions before computing cross products, fluxes, gradients, curls, or line integrals.
Energy method: Use potential or stored energy when forces are complicated but conservative structure is clear.
Numerical method: Use computation when analytic symmetry is insufficient, but verify the model against known limiting cases.

A correct formula used in the wrong geometry is still a wrong solution. State assumptions explicitly, such as point-like sources, infinite wires, ideal capacitors, negligible resistance, quasistatic behavior, or vacuum propagation.

Applications Across Science and Engineering

Electromagnetic concepts reappear throughout university study. In electrical engineering, circuit theory is the low-frequency limit of field theory, while high-frequency systems require transmission lines, waveguides, and antennas. In materials science, polarization, conductivity, magnetic response, and dielectric loss connect microscopic structure to macroscopic fields. In chemistry, molecular dipoles and spectroscopy depend on electromagnetic interactions. In biology and medicine, membrane potentials, electrophysiology, imaging, and radiation-matter interaction rely on electromagnetic models.

Modern devices often combine several principles at once. An electric motor uses magnetic force and torque; a generator uses induction; a transformer uses mutual inductance; a capacitor stores electric-field energy; an inductor stores magnetic-field energy; a radio link uses oscillating currents and electromagnetic radiation.

Interactive Tasks

Quiz: Test Your Knowledge

What is the electric field of a point charge proportional to as distance increases? (Inverse square of distance) (!Inverse distance) (!Distance squared) (!Distance cubed)

Which law is usually most efficient for finding the electric field of a highly symmetric charge distribution? (Gauss law) (!Faraday law) (!Lenz law) (!Ohm law)

What is the relationship between electrostatic electric field and electric potential? (Electric field is the negative gradient of potential) (!Electric field equals potential times charge) (!Electric field is the curl of potential) (!Electric field is independent of potential)

What quantity does a capacitor primarily store in its idealized field description? (Electric field energy) (!Magnetic monopoles) (!Mechanical momentum only) (!Gravitational potential)

What is the magnetic force on a moving positive charge proportional to? (Velocity cross magnetic field) (!Velocity dot magnetic field) (!Charge divided by magnetic field) (!Magnetic flux divided by time)

Which law relates magnetic field circulation to current in magnetostatics? (Ampere law) (!Coulomb law) (!Hooke law) (!Snell law)

What does the minus sign in Faraday law express? (Opposition to the change in magnetic flux) (!Creation of magnetic monopoles) (!Loss of electric charge) (!Negative resistance in every circuit)

Which term did Maxwell add to extend Ampere law to time varying electric fields? (Displacement current term) (!Coulomb friction term) (!Gravitational source term) (!Thermal diffusion term)

How are electric field magnetic field and propagation direction oriented in an ideal plane electromagnetic wave? (Mutually perpendicular) (!All parallel) (!Electric and magnetic fields parallel only) (!Magnetic field parallel to propagation)

What vector describes electromagnetic energy flow in classical field theory? (Poynting vector) (!Dipole vector) (!Drift vector) (!Gauss vector)

Memory Game

Coulomb law	Force between point charges follows an inverse square dependence
Gauss law	Electric flux through a closed surface depends on enclosed charge
Capacitance	Ratio of stored charge to potential difference
Lorentz force	Combined electric and magnetic force on a charged particle
Biot-Savart law	Magnetic field contribution produced by a current element
Faraday law	Changing magnetic flux produces an induced electromotive force
Inductance	Measure of flux linkage produced per current in a linear circuit
Poynting vector	Direction and rate of electromagnetic energy transport per area

Drag and Drop

Match the correct terms.	Topic
Enclosed electric charge	Gauss law source
Changing magnetic flux	Faraday law cause
Current and changing electric field	Ampere-Maxwell sources
Potential gradient	Electric field relation
Cross product of electric and magnetic fields	Electromagnetic energy flow direction

Match each physical phrase with the principle or field relationship shown in the second column.

Crossword Puzzle

Coulomb	Which scientist gives the name to the inverse square law for electrostatic force?
Gauss	Which law links closed surface electric flux to enclosed charge?
Ampere	Which name is associated with magnetic circulation around electric current?
Faraday	Which scientist is associated with electromagnetic induction?
Maxwell	Which physicist unified classical electricity magnetism and light in one field theory?
Inductance	What circuit property links changing current to an induced electromotive force?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Field Line Sketch: Draw electric field lines and equipotential lines for one positive charge, one negative charge, and a dipole. Explain where the field is strongest and how your drawing shows direction.
Capacitor Observation: Photograph or sketch three everyday devices that plausibly contain capacitors, research the capacitor's role in each device, and write a short comparison of energy storage, filtering, or timing functions.
Right-Hand Rule Video: Produce a two-minute video that demonstrates the right-hand rule for the magnetic field around a straight current-carrying wire and for magnetic force on a positive moving charge.
Electromagnetism Interview: Interview an engineering student, technician, researcher, or instructor about one practical use of electromagnetism and summarize the physical principle in your own words.

Standard

RC Circuit Experiment: Build or simulate an RC circuit, record capacitor voltage during charging or discharging, estimate the time constant from your data, and compare it with the theoretical value RC.
Magnetic Field Mapping: Use a compass, smartphone magnetometer, or laboratory probe to map the magnetic field near a bar magnet or coil, then discuss measurement limitations and background fields.
Faraday Induction Demonstration: Move a magnet through a coil connected to a sensitive meter or use a safe simulation, vary the speed and direction, and explain the observed emf using flux change and Lenz's law.
Science Museum Field Study: Visit a science museum, university laboratory open day, power facility visitor center, or technical exhibition and document at least three exhibits or systems that rely on electromagnetic principles.

Advanced

Numerical Electric Field Project: Write a computational model that calculates the electric potential and field of a nontrivial two-dimensional charge distribution, visualize the result, and validate it against a symmetric limiting case.
Maxwell Equation Case Study: Choose one technology such as a coaxial cable, waveguide, transformer, MRI system, or antenna and explain which Maxwell equations dominate the analysis at different scales or operating regimes.
Electromagnetic Energy Audit: Analyze energy transfer in a simple circuit or wave system using field energy and the Poynting vector, and compare the field viewpoint with the standard circuit-power viewpoint.
Research Communication Project: Produce a poster, article, podcast, or video for university peers that traces one concept from Coulomb's law through Maxwell's equations to a modern application, with equations, diagrams, assumptions, and properly credited sources.

Learning Assessment

Electrostatic Modeling Assessment: Derive the electric field on the symmetry axis of a uniformly charged ring, identify the far-field limit, and explain why that limit matches a point-charge model.
Potential and Energy Assessment: Compare two methods for finding the work required to assemble a small system of point charges, and justify when scalar potential is more efficient than direct force integration.
Magnetostatics Transfer Assessment: Design an Ampere-law argument for an ideal solenoid and then explain precisely why the same shortcut fails for a short finite solenoid.
Induction Reasoning Assessment: Given a conducting loop moving through a nonuniform magnetic field, predict the sign and qualitative time dependence of induced current before performing any algebra.
Maxwell Consistency Assessment: Explain why the original steady-current form of Ampere's law is incomplete for a charging capacitor and show how the displacement-current term resolves the inconsistency.
Wave Application Assessment: Starting from the source-free Maxwell equations, outline the derivation of an electromagnetic wave equation and connect the result to one real communication or sensing technology.

Evidence of Learning

Knowledge: You can explain electric charge, fields, potentials, flux, capacitance, current, magnetic force, induction, inductance, Maxwell's equations, and electromagnetic waves in a connected conceptual framework.
Mathematical skill: You can select and apply vector algebra, calculus, line integrals, surface integrals, differential operators, symmetry arguments, and differential equations appropriately.
Experimental and computational skill: You can design measurements or simulations, record data, estimate uncertainty, compare observations with theory, and identify the limits of idealized models.
Products: Your evidence may include solved derivations, field maps, circuit data, simulation code, laboratory reports, diagrams, presentations, posters, podcasts, or videos.
Transfer: You can use electromagnetic principles to explain or analyze unfamiliar systems such as sensors, motors, generators, antennas, biomedical devices, communication links, or energy-conversion equipment.