Leveraging Maxwell’s Equations for Accurate Simulation

Gauss's Law – The Foundation of Electric Fields

The foundation of our simulation rests upon Gauss’s Law, which describes the relationship between electric charges and the resultant electric field. Mathematically, this law is expressed as ∮ E ⋅ dA = Q / ε₀, where *E* represents the electric field vector, *dA* is a differential area element on a closed surface, *Q* is the enclosed charge, and ε₀ (approximately 8.854 × 10⁻¹² F/m) is the permittivity of free space. This equation dictates that the total flux of an electric field through any closed surface is proportional to the net charge enclosed within that surface.

Within the simulator, this law is implemented by calculating the electric field at every point in space and integrating it over a chosen Gaussian surface. The result directly determines the charge distribution needed to produce that specific field strength.

∮ E ⋅ dA = Q / ε₀

Coulomb’s Law – Forces Between Charges

Coulomb's Law quantifies the electrostatic force between two point charges. It states that the magnitude of this force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. The equation is F = k |q₁q₂| / r², where *F* is the force, *k* (approximately 8.9875 × 10⁹ N⋅m²/C²) is Coulomb’s constant, q₁ and q₂ are the magnitudes of the charges, and *r* is the distance between them.

The simulator utilizes this law to calculate forces between charged particles, allowing users to observe how changes in charge or separation affect the interaction.

F = k |q₁q₂| / r²

Ampère’s Law – Magnetic Fields Produced by Currents

Ampère's Law describes the magnetic field generated by electric currents. It states that the line integral of the magnetic field around a closed loop is proportional to the current passing through the area enclosed by the loop. Mathematically, this can be expressed as ∮ B ⋅ dl = μ₀I, where *B* represents the magnetic field vector, *dl* is an infinitesimal length element along the loop, μ₀ (4π × 10⁻⁷ T⋅m/A) is the permeability of free space, and *I* is the current flowing through the area.

Our simulation employs this law to model currents in wires and other conductive materials, accurately predicting the resulting magnetic field distribution.

∮ B ⋅ dl = μ₀I

Faraday’s Law – Induced Electromagnetic Fields

Faraday's Law describes how a changing magnetic field induces an electromotive force (EMF) in a circuit. This EMF, in turn, creates a current. The law is expressed as ε = -dΦB/dt, where *ε* represents the induced EMF, ΦB is the magnetic flux through the loop, and *t* is time. The negative sign indicates the direction of the induced EMF according to Lenz's Law.

Within the simulator, users can create dynamic scenarios involving changing magnetic fields, observing the resulting induced currents within simulated circuits.

ε = -dΦB/dt

Maxwell’s Equations – A Unified Framework

The simulation operates on Maxwell's complete set of equations, which unify electricity and magnetism. These include Gauss’s Law for Electricity, Gauss’s Law for Magnetism, Faraday’s Law of Induction, and Ampère-Maxwell’s Law. These equations provide a comprehensive description of electromagnetic phenomena, including the propagation of electromagnetic waves.

The simulator utilizes numerical methods – typically finite element analysis or finite difference schemes – to approximate solutions to these complex differential equations across the simulated space and time.

Time-Dependent Simulations

Beyond static fields, our platform allows for simulations of time-dependent electromagnetic phenomena. The equations are discretized in both space and time, allowing us to track how electric and magnetic fields evolve over time due to changing charge distributions or current flows. This is achieved through methods like the Finite Difference Time Domain (FDTD) method.

By adjusting parameters such as the rate of change of currents or charges, users can explore phenomena like electromagnetic wave propagation, transient responses in circuits, and even basic concepts like Lenz’s Law.

Frequently asked questions

Try it live

Everything above runs in your browser — open SPH Fluid and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.