Description: Title: Assignment 4Description:Derive an equation for Electromagneticwave equation in free space by giving atime varying equation for coupledelectric and magnetic field and they areresponsible for propagation ofelectromagnetic waves. use lect of week

According to faraday's law: ∇×E=-dBdt taking curl on both sides of the above equation.…

Answered: Description: Title: Assignment 4…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Description: Title: Assignment 4
Description:
Derive an equation for Electromagnetic
wave equation in free space by giving a
time varying equation for coupled
electric and magnetic field and they are
responsible for propagation of
electromagnetic waves. use lect of week

Expert Solution

Step 1

According to faraday's law:

$\nabla \times E = - \frac{d B}{d t}$

taking curl on both sides of the above equation.

$C u r l (\nabla \times E) = c u r l (- \frac{d B}{d t}) \nabla \times (\nabla \times E) = - \frac{d (\nabla \times B)}{d t} - - - - - (1) N o w, \nabla \times (\nabla \times E) = \nabla (\nabla . E) - \nabla^{2} E Using the above formula in equation 1, \Rightarrow \nabla (\nabla . E) - \nabla^{2} E = - \frac{d (\nabla \times B)}{d t} - - - - - - (2)$

Step 2

According to Gauss Law

$\nabla . D = ρ_{v} D = ε_{0} E \nabla . (ε_{0} E) = ρ_{v} \Rightarrow \nabla . E = \frac{ρ_{v}}{ε_{0}} \Rightarrow \nabla . E = \frac{ρ_{v}}{ε_{0}} For Free space, ρ_{v} = 0 ∴ \nabla . E = 0 Therefore from equation 2, \nabla (0) - \nabla^{2} E = - \frac{d (\nabla \times B)}{d t} - \nabla^{2} E = - \frac{d (\nabla \times B)}{d t} \nabla^{2} E = \frac{d (\nabla \times B)}{d t} \nabla^{2} E = μ_{0} \frac{d (\nabla \times H)}{d t} (∵ B = μ_{0} H)$

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