Final answer:
Maxwell's Equations in time-harmonic form for a lossy medium are represented with Gauss's Law for Electricity, Gauss's Law for Magnetism, Faraday's Law of Induction, and Ampere's Law with Maxwell's Addition. They describe how electric and magnetic fields behave in an oscillating manner due to a sinusoidal time dependence in a medium with attenuation.
Step-by-step explanation:
The four Maxwell's Equations describe the behavior of electric and magnetic fields and how they interact with matter. In the time-harmonic or phasor form, typically used when dealing with electromagnetic waves in a source-free (no free charges or currents) but lossy (attenuating) medium, the equations include sinusoidal time dependence expressed with the use of complex exponentials (phasors). Here are these equations in a lossy medium phasor form:
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- Gauss's Law for Electricity: \(\\abla \cdot \mathbf{D} = \rho_v\)
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- Gauss's Law for Magnetism: \(\\abla \cdot \mathbf{B} = 0\)
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- Faraday's Law of Induction: \(\\abla \times \mathbf{E} = -j\omega \mathbf{B}\)
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- Ampere's Law with Maxwell's Addition: \(\\abla \times \mathbf{H} = \mathbf{J} + j\omega \mathbf{D}\)
For a lossy medium, the electric field \(\mathbf{E}\) and magnetic field \(\mathbf{H}\) include the attenuation factor due to the medium's conductivity. The displacement current density \(\mathbf{D}\) is related to the electric field, and the current density \(\mathbf{J}\) to the charge carrier's motion in a material with a given conductivity. The symbol \( j \) is the imaginary unit, and \( \omega \) represents the angular frequency of the time-varying fields.