Interactive Lab
Simulations
Every simulation of the course in one place. They run in your browser, respond instantly to dragging and show the formula behind each number. Each one also appears in the section where it is needed.
Vector analysis
Vector Algebra
Drag two vectors and see their sum, difference, dot product (as a projection) and cross product (as an oriented area).
Open full-width →Gradient and Directional Derivative
Contour map of a scalar hill with its gradient field. Move a point and a direction to see the directional derivative.
Open full-width →Vector Field Explorer
Arrows, divergence and curl maps of 2-D vector fields, with a probe that shows net outflow and paddle-wheel rotation, and a loop that checks the divergence and Stokes theorems numerically.
- 2.2.3 Divergence
- 2.2.4 Curl
- 2.3.10 Differential operators in curvilinear coordinates
- 2.4.2 The divergence (Gauss's) theorem
- 2.4.3 Stokes' theorem
Coordinate Systems in 3-D
Rotate a 3-D scene, move a point in Cartesian, cylindrical or spherical coordinates and watch the local unit vectors and coordinate surfaces.
Open full-width →Electrostatics
Point-Charge Playground
Drag point charges, change their values and see the Coulomb forces, field vectors, field lines and equipotentials update instantly.
- 1.1 What electromagnetics is about
- 3.1.3 Coulomb's law
- 3.1.5 The superposition principle
- 3.1.7 Field lines
- 3.3.4 Equipotential surfaces
Building a Continuous Charge from Point Charges
Approximate a line, ring or disk charge by N point charges and watch the field at a point converge to the integral.
Open full-width →Finite Line Charge
Field of a finite line charge at a movable point, compared with the infinite-line result.
Open full-width →Ring, Disk and Annulus on Axis
On-axis field and potential of a charged ring, disk or annulus, with the point-charge and infinite-sheet limits.
- Problem 3.1.2: Circular loop source
- Problem 3.1.3: Circular disk source
- Home exercise 3.3.2: Circular disk source
- Home exercise 3.3.3: Annular disk source
Electron Beam Deflection
Accelerate an electron through a potential difference and deflect it between parallel plates onto a screen (CRT and electron gun).
Open full-width →Gaussian Surface Lab
Move and resize a Gaussian sphere among point charges. The numerically integrated flux always equals the enclosed charge over epsilon-zero.
Open full-width →Radial Field Profiles
D, E and V versus distance for spheres, shells, cylinders and coaxial lines, obtained from Gauss's law with adjustable radii and permittivities.
- Problem 3.2.5: Spherical shell with volume charge
- Home exercise 3.2.1: Uniformly charged sphere
- Home exercise 3.2.2: Nonuniformly charged sphere
- Home exercise 3.2.3: Uniformly charged cylinder
- Worked example 3.2.1: Air-spaced coaxial line
- Worked example 3.2.2: Charge inside a conducting shell
- Home exercise 3.4.3: Two-layer spherical capacitor
- Exercise 3.4.2: Point charge inside a dielectric shell
2-D Laplace / Poisson Field Solver
Finite-difference solver running in the browser: draw electrodes and dielectrics, solve for the potential and see equipotentials, field lines and capacitance.
- 3.2.5 Applications of Gauss's law
- 3.3.5 Laplace's and Poisson's equations
- Problem 3.3.2: Laplace's equation in only
- 3.4.8 The image principle
- Problem 3.4.1: The dielectric-filled parallel-plate capacitor
Work Along a Path
Line integral of a field along two different paths between the same points: path independence of conservative fields.
Open full-width →Poisson's Equation in a pn Junction
Depletion-approximation charge, field and potential of an abrupt pn junction with adjustable doping and bias.
Open full-width →Refraction of E at a Dielectric Interface
Boundary conditions in action: tangential E and normal D are continuous, so field lines bend at the interface.
Open full-width →Method of Images
A point or line charge above a grounded plane: field lines, the image charge, the induced surface charge and the attraction force.
Open full-width →Pushing a Dielectric Slab into a Capacitor
Drag the slab into the plates and see capacitance, stored energy, charge and the electrostatic force update, at constant voltage or constant charge.
Open full-width →Capacitor Designer
Parallel-plate, coaxial and spherical capacitors: capacitance, field maximum, breakdown voltage and stored energy.
Open full-width →Magnetostatics
Charged Particle in E and B
Drag the initial velocity of an electron or proton and watch it circle, drift in crossed fields or pass a velocity selector, with the forces qE and qv×B drawn on it.
- 4.1.1 The Lorentz force
- Home exercise 4.1.4: Charge in crossed fields
- Exercise 4.1.2: Electron in a uniform field
Hall Effect
Carriers in a current-carrying strip are pushed sideways by B until the Hall field balances them. Compare V_H = IB/nqt for metals and semiconductors and for electrons and holes.
Open full-width →Biot–Savart Field Lines
Field lines and a draggable probe for wires, a loop, a solenoid and a Helmholtz pair, with the numerically integrated Biot–Savart field checked against the closed forms.
- 4.1.3 Biot–Savart's law
- 4.1.7 Two fields every engineer should know
- Problem 4.1.2: Circular loop: the field on the axis
- Home exercise 4.1.2: Finite solenoid
Ampère's Law Profiles
Drag an Amperian circle through wires, tubes and a coaxial cable and see the enclosed current, H and B as functions of the radius.
- 4.1.8 Ampère's law at work
- Problem 4.1.4: Infinitely long thin wire (Ampère)
- Problem 4.1.5: Solid wire of radius b
- Problem 4.1.6: Hollow wire
- Problem 4.1.7: A given field: find the source
- Home exercise 4.1.5: Non-uniform current in a wire
- Home exercise 4.1.6: Coaxial cable
Forces Between Parallel Currents
Drag long parallel wires, set their currents in or out of the page, and see the field lines and the force per unit length, including the three-wire balance.
- Worked example 4.1.1: Parallel wires and the ampere
- 4.2.4 Mutual effect of current loops and inductance
- Problem 4.2.1: Force between two parallel currents
- Problem 4.2.2: Three parallel wires
DC Motor and Commutator
A rotating armature coil between magnet poles with its forces, the torque over one turn, the effect of the commutator, the mean torque and the power.
- 4.1.9 Applications
- Worked example 4.1.4: Force and torque on a DC-motor armature
- 4.2.8 Magnetic dipole and torque
- Problem 4.2.5: DC motor
Refraction of B at a Magnetic Interface
Drag the flux density across the interface of two magnetic media and see B_n and H_t stay continuous and the lines bend by tanθ₁/tanθ₂ = μ₁/μ₂.
Open full-width →Hysteresis Loop of a Ferromagnet
Drive soft and hard magnetic materials with a sinusoidal H and watch the initial curve, saturation, remanence, coercivity, minor loops and the hysteresis loss per cycle.
- 4.2.1 Magnetostatics in nonempty spaces
- 4.3.6 Saturation and hysteresis: the limits of the reluctance
Inductance and Magnetic Energy
L and the stored energy of a solenoid, a gapped toroid, a coaxial cable and two coupled coils, with a sketch, the formulas and a plot of L against the varied parameter.
- 4.2.4 Mutual effect of current loops and inductance
- 4.2.5 Magnetostatic energy
- Problem 4.2.3: The infinitely long solenoid
- Home exercise 4.2.1: Coaxial cable
- Home exercise 4.2.3: Energy stored in a toroid
- Worked example 4.2.1: Toroidal inductor and its air gap
Magnetic Circuit Solver
A gapped C-core and Problems 4.3.1–4.3.3 solved with the loop and node rules on the book's B–H curves, with the equivalent circuit, the mmf budget and the operating points.
- 4.3.4 Reluctances in series and in parallel
- 4.3.8 Fringing and leakage
- Problem 4.3.1: Two C-cores with two air gaps
- Problem 4.3.2: Three-limb core with two coils
- Problem 4.3.3: Series-parallel core with a gap in the centre limb
Lifting Electromagnet
A U-magnet over a steel plate: the gap induction, the pull B²S/μ0 on the two poles and its collapse as the plate is dragged away.
- 4.3.8 Fringing and leakage
- Home exercise 4.3.3: Lifting electromagnet
- Exercise 4.3.8: Lifting magnet at a given current
Maxwell's equations
Faraday's Law of Induction
A sliding bar, a loop in a changing field, a generator, a moving frame, the Faraday disk and a magnet, with Φ(t), e(t) = −dΦ/dt split into motional and transformer parts, and the induced current by Lenz's law.
- 5.3.3 The minus sign is Lenz's law
- 5.3.4 The two ways to change the flux
- Problem 5.3.1: Moving frame in a time-varying field
- Problem 5.3.2: The Faraday disk generator
- Home exercise 5.3.1: Circular loop in a changing field
- Home exercise 5.3.2: Sliding bar on rails
- Exercise 5.3.3: The rotating coil generator
Displacement Current
A capacitor driven by V0 cos ωt shows that Ampère's law gives the same circulation on the flat surface and on the bag through the gap only with ε∂E/∂t, plus the ratio σ/(ωε) of real media against frequency.
- 5.2.1 Why the term must be there: one loop, two surfaces
- 5.2.3 Conduction or displacement? The crossover frequency
- Home exercise 5.2.1: Displacement current in a capacitor
Uniform Plane Wave
A rotatable 3-D animation of E and H in a uniform plane wave, with the speed, wavelength, wave number, intrinsic impedance and Poynting vector for any f, εr and μr.
- Home exercise 5.1.1: A plane wave in free space
- Exercise 5.1.1: Plane wave in a dielectric
- 5.4.1 Twelve lectures, four equations