Consider the following equations with time factor e^jωt

1. \(\displaystyle\oint D_S.dS = \displaystyle\int \rho_{VS} dV\)

2. \(\displaystyle\oint E_S. dl = - j\omega B_S\)

3. \(\displaystyle\oint B_S.dS = 0\)

4. \(\displaystyle\oint H_S .dS = \displaystyle\int (J_S + j\omega D_S). dl\)

Which of the above Time-Harmonic Maxwell's equations are correct ?

Differential

Integral

Remarks

∇.D̅ = ρ_v

\(\oint_S \overline D . d \overline s = \int_V \rho_v dv\)

Gauss’ Law

∇.B̅ = 0

\(\oint_S \overline B . d \overline s =0\)

Nonexistence of isolated magnetic charge

\(\nabla x \overline E = - \frac{\partial \overline B}{\partial t}\)

\(\oint_L \overline E . d \overline l = - \frac{\partial }{\partial t} \int_s \overline B . d \overline s\)

Faraday’s Law

\(\nabla x \overline H =\overline J + \frac{\partial \overline D}{\partial t}\)

\(\oint_L \overline H . d \overline l = \int_s \left( \overline J + \frac{\partial \overline D}{\partial t} \right). d \overline s\)

Ampere’s circuital law