physics.maxwells_equations

Maxwell’s Equations Implementation

This module provides implementations of Maxwell’s four fundamental equations that describe the behavior of electric and magnetic fields in space and time.

The four equations are: 1. Gauss’s law for electricity: div(E) = rho/epsilon_0 2. Gauss’s law for magnetism: div(B) = 0 3. Faraday’s law of induction: curl(E) = -dB/dt 4. Ampère-Maxwell law: curl(B) = mu_0(J + epsilon_0*dE/dt)

Reference: https://en.wikipedia.org/wiki/Maxwell%27s_equations

Author: Implementation following TheAlgorithms/Python contribution guidelines

Attributes

SPEED_OF_LIGHT
VACUUM_PERMEABILITY
VACUUM_PERMITTIVITY
c

Functions

ampere_maxwell_law(→ bool)	Ampère-Maxwell law: curl(B) = mu_0(J + epsilon_0*dE/dt)
electromagnetic_wave_impedance(→ float)	Calculate the impedance of electromagnetic waves in a medium.
electromagnetic_wave_speed(→ float)	Calculate the speed of electromagnetic waves in a medium.
energy_density_electromagnetic(→ float)	Calculate the energy density of an electromagnetic field.
faraday_law(→ bool)	Faraday's law of electromagnetic induction: curl(E) = -dB/dt
gauss_law_electric(→ bool)	Gauss's law for electricity: div(E) = rho/epsilon_0
gauss_law_magnetic(→ bool)	Gauss's law for magnetism: div(B) = 0
poynting_vector_magnitude(→ float)	Calculate the magnitude of the Poynting vector (electromagnetic power flow).

Module Contents

physics.maxwells_equations.ampere_maxwell_law(magnetic_field_circulation: float, enclosed_current: float, electric_flux_change_rate: float, permeability: float = VACUUM_PERMEABILITY, permittivity: float = VACUUM_PERMITTIVITY) → bool

Ampère-Maxwell law: curl(B) = mu_0(J + epsilon_0*dE/dt)

In integral form: ∮B·dl = mu_0(I_enclosed + epsilon_0*dPhi_E/dt)

This law relates magnetic fields to electric currents and changing electric fields. Maxwell’s addition of the displacement current term (epsilon_0*dE/dt) was crucial for predicting electromagnetic waves.

Args:

magnetic_field_circulation: Line integral of B around closed loop (T·m) enclosed_current: Current passing through surface bounded by loop (A) electric_flux_change_rate: Rate of change of electric flux (V·m/s) permeability: Permeability of the medium (H/m), defaults to vacuum permittivity: Permittivity of the medium (F/m), defaults to vacuum

Returns:

bool: True if Ampère-Maxwell law is satisfied within numerical tolerance

Raises:

ValueError: If permeability or permittivity is non-positive

Example:

>>> ampere_maxwell_law(1.256e-6, 1.0, 0.0)
True
>>> ampere_maxwell_law(2.512e-6, 2.0, 0.0)
True
>>> ampere_maxwell_law(1.11e-5, 0.0, 1.0e12)
True

physics.maxwells_equations.electromagnetic_wave_impedance(permeability: float = VACUUM_PERMEABILITY, permittivity: float = VACUUM_PERMITTIVITY) → float

Calculate the impedance of electromagnetic waves in a medium.

The impedance Z_0 = sqrt(mu/epsilon) determines the ratio of electric to magnetic field strength in an electromagnetic wave.

Args:

permeability: Permeability of the medium (H/m) permittivity: Permittivity of the medium (F/m)

Returns:

float: Wave impedance of the medium (Ω - ohms)

Raises:

ValueError: If permeability or permittivity is non-positive

Example:

>>> abs(electromagnetic_wave_impedance() - 376.73) < 0.01
True
>>> impedance = electromagnetic_wave_impedance(
...     2*VACUUM_PERMEABILITY, VACUUM_PERMITTIVITY
... )
>>> abs(impedance - 532.0) < 1.0
True

physics.maxwells_equations.electromagnetic_wave_speed(permeability: float = VACUUM_PERMEABILITY, permittivity: float = VACUUM_PERMITTIVITY) → float

Calculate the speed of electromagnetic waves in a medium.

From Maxwell’s equations: c = 1/sqrt(mu_0*epsilon_0) in vacuum In a medium: v = 1/sqrt(mu*epsilon)

Args:

permeability: Permeability of the medium (H/m) permittivity: Permittivity of the medium (F/m)

Returns:

float: Speed of electromagnetic waves in the medium (m/s)

Raises:

ValueError: If permeability or permittivity is non-positive

Example:

>>> abs(electromagnetic_wave_speed() - 2.998e8) < 1e6
True
>>> speed = electromagnetic_wave_speed(
...     VACUUM_PERMEABILITY, 2*VACUUM_PERMITTIVITY
... )
>>> abs(speed - 2.12e8) < 1e7
True

physics.maxwells_equations.energy_density_electromagnetic(electric_field: float, magnetic_field: float, permittivity: float = VACUUM_PERMITTIVITY, permeability: float = VACUUM_PERMEABILITY) → float

Calculate the energy density of an electromagnetic field.

The energy density u = (1/2)*(epsilon_0*E^2 + B^2/mu_0) represents the electromagnetic energy stored per unit volume.

Args:

electric_field: Magnitude of electric field (V/m) magnetic_field: Magnitude of magnetic field (T) permittivity: Permittivity of the medium (F/m) permeability: Permeability of the medium (H/m)

Returns:

float: Energy density (J/m³)

Raises:

ValueError: If permittivity or permeability is non-positive

Example:

>>> abs(energy_density_electromagnetic(1000, 1e-3) - 0.398) < 0.001
True
>>> abs(energy_density_electromagnetic(0, 1.0) - 397887) < 1
True
>>> abs(energy_density_electromagnetic(377, 1e-6) - 1.0e-6) < 1e-6
True

physics.maxwells_equations.faraday_law(electric_field_circulation: float, magnetic_flux_change_rate: float) → bool

Faraday’s law of electromagnetic induction: curl(E) = -dB/dt

In integral form: ∮E·dl = -dPhi_B/dt

This law describes how a changing magnetic field induces an electric field. The induced electric field opposes the change in magnetic flux (Lenz’s law).

Args:

electric_field_circulation: Line integral of E around closed loop (V) magnetic_flux_change_rate: Rate of change of magnetic flux (Wb/s or V)

Returns:

bool: True if Faraday’s law is satisfied within numerical tolerance

Example:

>>> faraday_law(10.0, -10.0)
True
>>> faraday_law(-5.0, 5.0)
True
>>> faraday_law(0.0, 0.0)
True
>>> faraday_law(10.0, 10.0)
False

physics.maxwells_equations.gauss_law_electric(electric_field_magnitude: float, surface_area: float, enclosed_charge: float, permittivity: float = VACUUM_PERMITTIVITY) → bool

Gauss’s law for electricity: div(E) = rho/epsilon_0

In integral form: ∮E·dA = Q_enclosed/epsilon_0

This law states that the electric flux through any closed surface is proportional to the total electric charge enclosed within that surface.

Args:

electric_field_magnitude: Magnitude of electric field (V/m or N/C) surface_area: Area of the closed surface (m²) enclosed_charge: Total charge enclosed by the surface (C - coulombs) permittivity: Permittivity of the medium (F/m), defaults to vacuum

Returns:

bool: True if Gauss’s law is satisfied within numerical tolerance

Raises:

ValueError: If surface_area is negative or permittivity is non-positive

Example:

>>> gauss_law_electric(1000, 1.0, 8.854e-9)
True
>>> gauss_law_electric(500, 2.0, 8.854e-9)
True
>>> gauss_law_electric(-100, 1.0, 8.854e-9)
False

physics.maxwells_equations.gauss_law_magnetic(surface_area: float) → bool

Gauss’s law for magnetism: div(B) = 0

In integral form: ∮B·dA = 0

This law states that there are no magnetic monopoles - the magnetic flux through any closed surface is always zero. Magnetic field lines always form closed loops or extend to infinity.

Args:

surface_area: Area of the closed surface (m²)

Returns:

bool: Always True for physically realistic magnetic fields,

False if net flux is non-zero (indicating monopoles)

Raises:

ValueError: If surface_area is negative

Example:

>>> gauss_law_magnetic(2.0)
True
>>> gauss_law_magnetic(0.0)
True
>>> gauss_law_magnetic(5.0)
True

physics.maxwells_equations.poynting_vector_magnitude(electric_field: float, magnetic_field: float, permeability: float = VACUUM_PERMEABILITY) → float

Calculate the magnitude of the Poynting vector (electromagnetic power flow).

The Poynting vector S = (1/mu_0) * E x B represents the directional energy flux density of an electromagnetic field (power per unit area).

Args:

electric_field: Magnitude of electric field (V/m) magnetic_field: Magnitude of magnetic field (T) permeability: Permeability of the medium (H/m)

Returns:

float: Magnitude of Poynting vector (W/m²)

Raises:

ValueError: If permeability is non-positive

Example:

>>> abs(poynting_vector_magnitude(1000, 1e-6) - 795.8) < 1.0
True
>>> abs(poynting_vector_magnitude(377, 1.0) - 3.0e8) < 1e6
True
>>> poynting_vector_magnitude(0, 1.0)
0.0

physics.maxwells_equations.SPEED_OF_LIGHT = 299792458

physics.maxwells_equations.VACUUM_PERMEABILITY = 1.2566370614359173e-06

physics.maxwells_equations.VACUUM_PERMITTIVITY = 8.8541878128e-12

physics.maxwells_equations.c