Electromagnetic Waves

What Are Electromagnetic Waves?

Electromagnetic waves are self-sustaining oscillations of electric and magnetic fields that travel through space at the speed of light. They are produced by accelerating charges and do not require a medium — they can propagate through vacuum.

The key idea: a changing electric field generates a magnetic field, and a changing magnetic field generates an electric field. This mutual induction allows the wave to move forward.

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Derivation from Maxwell’s Equations (Conceptual)

In a region with no charges ($\rho = 0$) and no currents ($\mathbf{J} = 0$), Maxwell’s equations in vacuum become:

• $\nabla \cdot \mathbf{E} = 0$
• $\nabla \cdot \mathbf{B} = 0$
• $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
• $\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Taking the curl of Faraday’s law and using the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$, with $\nabla \cdot \mathbf{E} = 0$, we get:

This is the wave equation for $\mathbf{E}$. A similar equation holds for $\mathbf{B}$:

Comparing with the standard wave equation $\nabla^2 f = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$, the wave speed is:

where:

• $\mu_0 = 4\pi \times 10^{-7} \, \text{N/A}^2$ (permeability of free space)
• $\varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2$ (permittivity of free space)

Plugging in the numbers gives $c \approx 3.00 \times 10^8 \, \text{m/s}$ — the speed of light in vacuum.

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Properties of Electromagnetic Waves

• Transverse nature: Both $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to the direction of propagation.
• Mutually perpendicular: $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to each other.
• In phase: The electric and magnetic fields oscillate in phase — they reach maxima and minima together.
• Relation between field magnitudes: At any point and time,

where $E$ and $B$ are the instantaneous magnitudes of the electric and magnetic fields.

• Energy transport: Electromagnetic waves carry energy. The energy per unit volume (energy density) is: …