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## Main Question or Discussion Point

In a chapter building up to the theory of plane waves my book starts by introducing

time harmonic electric fields and defines a special case of Gauss's law.

curl(H) = J + dD/dt

curl(H) = sigma * E + epsilon * dE/dt

if E is time harmonic and spacially dependent... E(x,y,z,t) let E' represent the phasor form

curl(H) = sigma * E' + epsilon * j * w * E'

curl(H) = (sigma + epsilon*j*w) E'

of curl(H) = jw(epsilon - j*sigma/w) E'

where epsilon - j*sigma/w = epsilon_c (complex permittivity)

given that... divergence(curl(H)) = 0....

divergence( jw * epsilon_c * E') = 0

therefore divergence(E) = 0

so pv (volume charge density) = 0 by Gauss's law

I am very confused why a time harmonic E field can never bound a charge source and why it's divergence is always zero as my book seems to suggest.

I am guessing of have missed a major assumption and or am misinterpreting something? Looking for some guidance. Thanks!

time harmonic electric fields and defines a special case of Gauss's law.

curl(H) = J + dD/dt

curl(H) = sigma * E + epsilon * dE/dt

if E is time harmonic and spacially dependent... E(x,y,z,t) let E' represent the phasor form

curl(H) = sigma * E' + epsilon * j * w * E'

curl(H) = (sigma + epsilon*j*w) E'

of curl(H) = jw(epsilon - j*sigma/w) E'

where epsilon - j*sigma/w = epsilon_c (complex permittivity)

given that... divergence(curl(H)) = 0....

divergence( jw * epsilon_c * E') = 0

therefore divergence(E) = 0

so pv (volume charge density) = 0 by Gauss's law

I am very confused why a time harmonic E field can never bound a charge source and why it's divergence is always zero as my book seems to suggest.

I am guessing of have missed a major assumption and or am misinterpreting something? Looking for some guidance. Thanks!