What does the epsilon_r slider control?

It sets the relative permittivity of the next dielectric region you paint, or of the Prism and Glass lens presets. A higher value slows the wave more and bends it more sharply at the boundary.

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🌈 2D EM Wave FDTD Simulator (Dielectrics & Conductors)

Interactive 2D electromagnetic wave simulation using the Finite Difference Time Domain (FDTD) method on a Yee grid, with real dielectric and lossy-conductor regions producing genuine refraction, reflection and absorption.

Electromagnetism2DAdvanced60 FPS📱 Mobile-adapted⇄ 3D version

Details

CategoryElectromagnetism

RenderingCanvas 2D

DifficultyAdvanced

Target FPS60

ControlsFrequency and amplitude sliders shape the oscillating source; two buttons switch between a point source and a left-edge plane wave. A material toolbar paints PEC mirror, dielectric (with an epsilon-r slider), lossy conductor or erase by click-drag on the canvas with an adjustable brush size. Five presets instantly build a double slit, a refracting prism, a plano-convex glass lens or an absorbing wall. Field-range and speed sliders adjust the color scale and steps-per-frame; pause and reset buttons control playback.

FDTDMaxwell equationsYee griddielectric refractionSnell's lawwave absorptionelectromagnetism2d

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What this demonstrates

The FDTD method (Yee, 1966) solves Maxwell's curl equations on a staggered spatial grid with leapfrog time-stepping. Here each grid cell also carries a real relative permittivity ε_r and conductivity σ, updated with the standard lossy-medium FDTD equation C_aE + C_bcurlH. A perfect electric conductor (PEC) forces E_z=0 and reflects the wave totally; a dielectric slows the local wave speed by 1/√ε_r, bending wavefronts at its boundary exactly as Snell's law predicts; a lossy metal has finite conductivity and dissipates the field's energy as it passes through, producing genuine absorption rather than a hard wall.

How to use

Point source: single oscillating dipole radiating circular wavefronts
Plane wave: coherent wave injected from the left edge
Pick a Material tool and drag on the canvas to paint that region; erase to clear it
Raise the dielectric's ε_r slider to bend light more sharply, just like a higher-index glass
Use Presets for an instant prism, plano-convex glass lens, double slit or absorbing wall

Did you know?

Real FDTD solvers assign every cell its own ε_r, μ_r and σ this same way to model glass fibres, human tissue, or radar absorbing paint — the update equation here is the textbook lossy-medium form, just run on a small 2D grid you can paint on directly instead of importing a CAD model.

About the 2D FDTD Simulator with Real Dielectrics and Conductors

This simulator solves Maxwell's curl equations on a 160×120 Yee grid with the Finite Difference Time Domain (FDTD) method, running the transverse-magnetic (TM_z) mode: E_z, H_x and H_y evolve by leapfrog time-stepping. Unlike a pure perfect-conductor sandbox, every cell here carries its own relative permittivity ε_r and conductivity σ, updated with the standard lossy-medium FDTD equation E_z = C_aE_z + C_b·curlH. That single change is what lets painted glass genuinely refract the wave and painted metal genuinely absorb it, instead of just blocking it.

Pick a material with the Material tool — PEC forces E_z=0 and reflects totally, Dielectric slows the local wave speed by 1/√ε_r and bends wavefronts at its boundary the way real glass bends light, and Lossy metal carries a finite conductivity that dissipates the field's energy as it crosses. The Prism and Glass lens presets fill real triangular and lens-shaped dielectric regions so you can watch genuine Snell's-law refraction and focusing, not a drawn outline.

Frequently Asked Questions

How is this different from a simple reflect-only FDTD demo?

Most simple demos only support perfect electric conductors (PEC), which just block or bounce the wave. This simulator additionally tracks a real permittivity and conductivity per cell and runs the standard lossy-dielectric FDTD update, so it reproduces refraction (bending at a material boundary) and absorption (energy loss inside a lossy conductor), not just reflection.

What is happening physically when a wave enters the dielectric?

Inside a dielectric with relative permittivity ε_r, the wave travels at c/√ε_r instead of c. Because the wavefront crosses the boundary at an angle and slows down on one side, it bends — exactly the mechanism behind Snell's law and why the prism and lens presets focus and split the wave the way real glass would.

What is the lossy-medium update equation?

Each cell updates as E_z = C_aE_z + C_b·curl(H), where C_a and C_b depend on that cell's ε_r and σ. C_a < 1 when σ > 0, so energy decays as the wave crosses a lossy conductor; C_b shrinks with higher ε_r, which is what slows the wave inside a dielectric.

What do PEC, Dielectric and Lossy metal actually paint?

PEC paints a perfect electric conductor: E_z is forced to zero every step, so the wave reflects totally with no loss. Dielectric paints a region with the ε_r you set on the slider (glass-like, no conductivity), which refracts and slows the wave but does not absorb it. Lossy metal paints a region with finite conductivity, which absorbs energy from the wave as it passes, similar to radar-absorbing material.

What does the ε_r slider control?

ε_r is the relative permittivity of the next dielectric region you paint (or of the Prism/Glass lens presets). Higher ε_r slows the wave more inside that material and bends it more sharply at the boundary, the same way a higher refractive-index glass bends light more than window glass.

Why don't waves bounce off the edges of the canvas?

A 10-cell absorbing layer rings the grid and gently damps the field as it approaches the boundary, mimicking an open, infinite domain so outgoing waves leave cleanly instead of reflecting back and contaminating the interior.

What does the Courant value mean?

Courant refers to the Courant–Friedrichs–Lewy stability condition. The time step must satisfy c·Δt/Δx ≤ 1/√2 for the vacuum wave speed; since a dielectric only slows the wave further, this fixed ratio stays stable everywhere on the grid, painted or not.

Is this physically accurate?

The update equations are the textbook lossy-medium Yee-grid FDTD scheme, so the refraction, reflection and absorption you see are governed by the same physics as a professional solver. It uses normalised units and a simple absorbing layer rather than a full perfectly matched layer, so it is qualitatively accurate and ideal for learning rather than for precision antenna design.

Where is this kind of simulation used in the real world?

Engineers use FDTD with spatially-varying permittivity and conductivity to design optical fibres and photonic waveguides, model radar-absorbing coatings on aircraft, and simulate how radio waves are absorbed by human tissue for safety standards — all the same physics you are painting here, just on a much larger 3D grid.

⚙ Under the hood

A 2D Yee-grid FDTD solver for Maxwell's curl equations (TMz mode) where painted regions carry a real per-cell permittivity and conductivity, so glass genuinely refracts the wave via Snell's law and lossy metal genuinely absorbs it, alongside classic PEC reflection.

FDTDMaxwell equationsYee griddielectric refractionSnell's lawwave absorptionelectromagnetism2d

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install