Closed path in a high-frequency electromagnetic field. A high-frequency time-harmonic electromagnetic field exists in a lossy medium. The current density vector in the medium is J, the electric flux density vector is D, and the magnetic flux density vector is B. The line integral of the magnetic field intensity vector, H, along an arbitrary planar contour (closed path) in the medium equals the flux of the following vector through a flat surface spanned over the contour:

Explanation:

The concept of amperes law is applied;

Given the current density vector in the medium is J,

the electric flux density vector is D,

and the magnetic flux density vector is B. The line integral of the magnetic field intensity H

considering the orientation of the contour is adopted according to ampere law;

deta x B = Jc + JD

delta x B = Jc + partial differential D/partial differential t

where Jc = conduction current density

JD = displacement current density

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explanation: