Poynting Vector Calculator
Poynting Vector Calculator computes power flux of an EM wave, visualizing energy flow for electromagnetism studies.
Formulas Used in Poynting Vector Calculator
The calculator uses the following formulas for a plane wave in free space:
Poynting Vector:
\\[ \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \\]Plane Wave Fields:
\\[ \mathbf{E} = E_0 \cos(kz – \omega t) \hat{x} \\] \\[ \mathbf{B} = \frac{E_0}{c} \cos(kz – \omega t) \hat{y} \\]Poynting Vector Magnitude:
\\[ |\mathbf{S}| = \frac{E_0^2}{\mu_0 c} \cos^2(kz – \omega t) \\]Wave Parameters:
\\[ k = \frac{2\pi}{\lambda}, \quad \omega = kc, \quad c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \\]Where:
- \\( \mathbf{S} \\): Poynting vector (W/m²)
- \\( \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 = 0 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}
\\]
\\[
|\mathbf{S}| = \frac{100^2}{4\pi \times 10^{-7} \cdot 2.99792458 \times 10^8} \cdot \cos^2(628.318 \cdot 0 – 1.884 \times 10^{11} \cdot 0) \approx 26525.8 \, \text{W/m}^2
\\]
Result: |S| = 26525.8 W/m² at t = 0 s
Example 2: Higher Amplitude
Input: E₀ = 200 V/m, λ = 0.01 m, z = 0 m, t = 0 s
\\[
|\mathbf{S}| = \frac{200^2}{4\pi \times 10^{-7} \cdot 2.99792458 \times 10^8} \cdot \cos^2(0) \approx 106103.3 \, \text{W/m}^2
\\]
Result: |S| = 106103.3 W/m² at t = 0 s
Example 3: Different Wavelength
Input: E₀ = 100 V/m, λ = 0.05 m, z = 0 m, t = 0 s
\\[
k = \frac{2\pi}{0.05} \approx 125.664 \, \text{rad/m}
\\]
\\[
\omega \approx 3.769 \times 10^{10} \, \text{rad/s}
\\]
\\[
|\mathbf{S}| = \frac{100^2}{4\pi \times 10^{-7} \cdot 2.99792458 \times 10^8} \cdot \cos^2(0) \approx 26525.8 \, \text{W/m}^2
\\]
Result: |S| = 26525.8 W/m² at t = 0 s