Maxwell’s Equations Solver
Maxwell’s Equations Solver computes electric and magnetic fields of a plane wave in free space, visualizing their oscillations for electromagnetism studies.
Formulas Used in Maxwell’s Equations Solver
The solver uses the following formulas for a plane wave in free space:
Maxwell’s Equations:
\\[ \nabla \cdot \mathbf{E} = 0, \quad \nabla \cdot \mathbf{B} = 0 \\] \\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\]Plane Wave Solutions:
\\[ \mathbf{E} = E_0 \cos(kz – \omega t) \hat{x} \\] \\[ \mathbf{B} = \frac{E_0}{c} \cos(kz – \omega t) \hat{y} \\]Wave Parameters:
\\[ k = \frac{2\pi}{\lambda}, \quad \omega = kc, \quad c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \\]Where:
- \\( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \\): Permeability of free space
- \\( \epsilon_0 = 8.854187817 \times 10^{-12} \, \text{F/m} \\): Permittivity of free space
- \\( c \approx 2.99792458 \times 10^8 \, \text{m/s} \\): Speed of light
- \\( E_0 \\): Electric field amplitude (V/m)
- \\( \lambda \\): Wavelength (m)
- \\( k \\): Wave number (rad/m)
- \\( \omega \\): Angular frequency (rad/s)
- \\( z \\): Position (m)
- \\( t \\): Time (s)
- \\( \mathbf{E}, \mathbf{B} \\): Electric and magnetic fields
Example Calculations
Example 1: Standard Case
Input: E₀ = 100 V/m, λ = 0.01 m, z = 0 m, t = 1e-10 s
\\[
k = \frac{2\pi}{0.01} \approx 628.318 \, \text{rad/m}
\\]
\\[
\omega = 628.318 \cdot 2.99792458 \times 10^8 \approx 1.884 \times 10^{11} \, \text{rad/s}
\\]
\\[
E_x = 100 \cdot \cos(628.318 \cdot 0 – 1.884 \times 10^{11} \cdot 1 \times 10^{-10}) \approx 80.901 \, \text{V/m}
\\]
\\[
B_y = \frac{100}{2.99792458 \times 10^8} \cdot \cos(628.318 \cdot 0 – 1.884 \times 10^{11} \cdot 1 \times 10^{-10}) \approx 2.699 \times 10^{-7} \, \text{T}
\\]
Result: E_x = 80.901 V/m, B_y = 2.699e-7 T
Example 2: Higher Amplitude
Input: E₀ = 500 V/m, λ = 0.01 m, z = 0 m, t = 1e-10 s
\\[
E_x \approx 404.508 \, \text{V/m}, \quad B_y \approx 1.349 \times 10^{-6} \, \text{T}
\\]
Result: E_x = 404.508 V/m, B_y = 1.349e-6 T
Example 3: Different Wavelength
Input: E₀ = 100 V/m, λ = 0.05 m, z = 0 m, t = 1e-10 s
\\[
k = \frac{2\pi}{0.05} \approx 125.664 \, \text{rad/m}
\\]
\\[
\omega \approx 3.769 \times 10^{10} \, \text{rad/s}
\\]
\\[
E_x \approx 99.294 \, \text{V/m}, \quad B_y \approx 3.311 \times 10^{-7} \, \text{T}
\\]
Result: E_x = 99.294 V/m, B_y = 3.311e-7 T