Relation Between Electric Field, Magnetic Field, and Speed of Light

In electromagnetic waves, the electric field (EE), magnetic field (BB), and speed of light (cc) are related through Maxwell’s equations.

Mathematical Relation:

c=EBc = \frac{E}{B}

where:

• cc = Speed of light in vacuum (3.0×1083.0 \times 10^8 m/s)
• EE = Electric field strength (V/m)
• BB = Magnetic field strength (T, Tesla)

This equation shows that the ratio of the magnitudes of electric field and magnetic field in an electromagnetic wave gives the speed of light.

Derivation from Maxwell’s Equations

Maxwell’s equations predict the existence of electromagnetic waves, where the electric and magnetic fields oscillate perpendicular to each other and to the direction of wave propagation.

1. From Faraday’s Law:

   ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}This shows that a changing magnetic field (B\mathbf{B}) induces an electric field (E\mathbf{E}).

2. From Ampère’s Law (with Maxwell’s Correction):

   ∇×B=μ0ε0∂E∂t\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}This shows that a changing electric field (E\mathbf{E}) induces a magnetic field (B\mathbf{B}).

3. Wave Equation for Electromagnetic Waves:

   c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}where:

   • μ0\mu_0 = Permeability of free space (4π×10−7 H/m4\pi \times 10^{-7} \, \text{H/m})
   • ε0\varepsilon_0 = Permittivity of free space (8.85×10−12 F/m8.85 \times 10^{-12} \, \text{F/m})

   Substituting values:

   c=1(4π×10−7)(8.85×10−12)c = \frac{1}{\sqrt{(4\pi \times 10^{-7}) (8.85 \times 10^{-12})}} c≈3.0×108 m/sc \approx 3.0 \times 10^8 \text{ m/s}

Key Takeaways:

1. Electromagnetic waves travel at the speed of light, given by c=EBc = \frac{E}{B}.
2. The speed of light depends on the permittivity and permeability of free space: c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}.
3. The electric field (EE) and magnetic field (BB) are perpendicular to each other and to the direction of wave propagation.

This fundamental relation is crucial in understanding electromagnetic wave propagation, optics, and relativistic physics.