Maxwell's Equations in Vacuum and Electromagnetic Wave Propagation

UPSC CSE / IFoS Paper I — Electricity and Magnetism

1. Maxwell's Field Equations in Free Space

In vacuum free of electric charges (\rho = 0) and conduction currents (\mathbf{J} = 0), Maxwell's equations in SI units are:

1. Gauss's Law for Electricity: \nabla \cdot \mathbf{E} = 0

2. Gauss's Law for Magnetism: \nabla \cdot \mathbf{B} = 0

3. Faraday's Law of Induction: \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

4. Ampère-Maxwell Circuital Law: \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}

2. Derivation of the Wave Equation for \mathbf{E}

Taking the curl of Faraday's Law: \nabla \times (\nabla \times \mathbf{E}) = -\nabla \times \left(\frac{\partial \mathbf{B}}{\partial t}\right)

Using the standard vector differential identity \nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}: \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B})

Since \nabla \cdot \mathbf{E} = 0, this simplifies to: -\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t}\left(\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)

Rearranging gives the classical three-dimensional vector wave equation: \nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0

Comparing this with the standard 3D wave equation \nabla^2 \psi - \frac{1}{v^2}\frac{\partial^2 \psi}{\partial t^2} = 0, we conclude that electromagnetic perturbations propagate as waves with phase speed: c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 2.9979 \times 10^8 \text{ m/s}

3. The Poynting Vector and Energy Flux

The instantaneous rate of energy flow per unit area is given by Poynting's vector: \mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})

For a plane harmonic wave propagating in the +z direction: \langle S \rangle = \frac{1}{2} \epsilon_0 c E_0^2 = \frac{E_0^2}{2 \mu_0 c}