Using Maxwell’s equations in free space, derive the wave equatio

Question

Using Maxwell’s equations in free space, derive the wave equation for the electric and magnetic field vectors. (5+5) c) The expression of the electric field associated with an electromagnetic wave in vacuum is given by

$\vec{E}=(800 Vm^{-1})$ $\hat{x}$ sin(2π x 10^{8 _t+kz)}

Determine the wave number, frequency, the direction of propagation and the magnitude and direction of the magnetic field associated with the wave.

22 Jan 2025

Answer :

Word Count : 645

### Part 1: Deriving the Wave Equation from Maxwell's Equations To derive the wave equation for the electric and magnetic fields, let's start with Maxwell's equations in free space: 1. Gauss's Law for Electricity: \[ \nabla \cdot \vec{E} = 0 \] This implies there are no free charges in the region. 2. Gauss's Law for Magnetism: \[ \nabla \cdot \vec{B} = 0 \] This states that there are no magnetic monopoles in the region. 3. Faraday's Law of Induction: \[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \] 4. Ampère's Law (without current): \[ \nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \] where $\mu_0$ is the permeability of free space, and $\epsilon_0$ is the permittivity of free space. By taking the curl of both sides of Faraday's law: \[ \nabla \times (\nabla \times \vec{E}) = -\nabla \times \frac{\partial \vec{B}}{\partial t} \] Using Ampère's law to replace $\nabla \times \vec{B}$, we get: \[ \nabla \times (\nabla \times \vec{E}) = -\mu_0 ___ __________ ___ ______ ___ _______ ____ __________ _____.
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