Learn Maxwell's Equations with interactive simulations. Understand electromagnetic field theory, wave equations, and applications with step-by-step visualizations and real-world examples.
Maxwell's equations are a set of four fundamental partial differential equations that describe the behavior of electric and magnetic fields and their interactions with matter. Formulated by James Clerk Maxwell in the 1860s, these equations unified electricity, magnetism, and optics into a single theoretical framework and predicted the existence of electromagnetic waves.
These equations form the foundation of classical electromagnetism, classical optics, and electric circuits. They demonstrate that electric and magnetic fields propagate at the speed of light as electromagnetic waves, thus unifying electromagnetic theory with optics.
The mathematical foundation of electromagnetic theory
The electric flux through any closed surface is proportional to the enclosed electric charge.
∇ · E = ρ/ε₀
∮S E · dA = Qenc/ε₀
The net magnetic flux through any closed surface is zero, indicating no magnetic monopoles exist.
∇ · B = 0
∮S B · dA = 0
A changing magnetic field creates an electric field (electromagnetic induction).
∇ × E = -∂B/∂t
∮C E · dl = -dΦB/dt
Magnetic fields are generated by electric currents and changing electric fields (displacement current).
∇ × B = μ₀J + μ₀ε₀∂E/∂t
∮C B · dl = μ₀Ienc + μ₀ε₀dΦE/dt
Maxwell's equations have numerous practical applications across science and engineering:
Radio, TV, cellular networks, WiFi, and Bluetooth operate on electromagnetic wave principles.
Understanding light propagation, lenses, mirrors, and optical instruments.
Design of motors, generators, transformers, and electrical circuits.
MRI machines use strong magnetic fields and radio waves for imaging.
Detection and ranging using reflected electromagnetic waves.
Solar panels and wind turbines based on electromagnetic principles.
Perform calculations for various electromagnetic phenomena
Calculate electric field from point charges
Electric Field Strength: 0 N/C
Formula: E = kQ/r²
Calculate magnetic field from current-carrying wire
Magnetic Field Strength: 0 T
Formula: B = μ₀I/(2πr)
Calculate properties of electromagnetic waves
kpi: 0 m
Energy: 0 J
Speed: 299,792,458 m/s
Calculate capacitance and energy stored
Capacitance: 0 F
Energy Stored: 0 J
Charge: 0 C
Convert between different units used in electromagnetism
Explore electromagnetic phenomena through hands-on visualizations
Visualize electric field lines around point charges
Step 1: Electric field lines radiate outward from positive charges and inward toward negative charges.
Observe propagating electromagnetic waves with oscillating electric and magnetic fields
Step 1: Stationary wave with electric and magnetic fields perpendicular to each other and propagation direction.
Interactive three-dimensional visualization of electromagnetic fields and wave propagation
3D Simulation Visualization
Interactive 3D model showing electromagnetic field lines and wave propagation
3D representation of electric field vectors around charges
Animated visualization of magnetic field lines around currents
Adjust parameters like charge, current, and field strength