Wave equation in inhomogeneous media -- Question about the formula

AI Thread Summary
The discussion centers on the derivation of the wave equation in inhomogeneous media, particularly focusing on the term ε0/ε. It begins with the equation ∇ × H = ∂D/∂t and substitutes D with ε(r)E, leading to the modified equation involving E. The transformation further simplifies to ∇ × [1/ε(r)(∇ × H)] = -μo ∂²H/∂t², incorporating the relationship B = μoH. Multiplying by ε0 reveals the connection between ε0, μo, and the speed of light, emphasizing the importance of these constants in wave propagation. This clarification aids in understanding the role of permittivity in the wave equation.
macabre
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Well, the first equation is,

##\nabla \times H = \frac{\partial D}{\partial t}##

to which we note, ##D = \epsilon(r)E## to get,

##\frac{1}{\epsilon(r)}\nabla \times H = \frac{\partial E}{\partial t}##

so,

##\nabla\times[\frac{1}{\epsilon(r)}(\nabla\times H)] = -\mu_o \frac{\partial^2 H}{\partial^2 t}##

where we've snuck in ##B=\mu_o H##. Multiply both sides by ##\epsilon_o## and note that ##\epsilon_o \mu_o = \frac{1}{c^2}##
 
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