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As learning laser fundumentals, I've just reviewed the boundary conditions for electromagnetic waves.

However, I came back to a point that confused me in the past and wanna get it clear now :)

One of the boundary conditions, regarding the magnetic fields parallel to the medium-interface is

[itex]H_{1//}-H_{2//}=J_s[/itex]

where [itex]H_{1//} \; and \; H_{2//} \; \text{are magnetic field strengths and} \; J_s \; \text{is the electric current per unit area} [/itex].

I found the derivation in my text book

[itex]\oint_{C} \vec{H} \cdot d\vec{l} = \int_{S} \vec{J} \cdot d\vec{s} + \frac{d \int_{S} \vec{D} \cdot d\vec{s}}{d \, t}[/itex]

Note that [itex]bc \approx 0 \; and \; da \approx 0[/itex] so that flux of electric fields is ignored, hence

[itex]ab \cdot H_{1//}+bc \cdot H_{\perp}+ cd \cdot H_{2//}+ da \cdot H_{\perp}= J_s \cdot AREA_{abcd}[/itex]

where [itex]ab=-cd, \; bc=-da[/itex], thus

[itex]ab \cdot H_{1//}- ab \cdot H_{2//} = J_s \cdot AREA_{abcd}[/itex]

and now, I just can't get [itex]H_{1//}-H_{2//}=J_s[/itex] because [itex]\frac{AREA_{abcd}}{ab} \not= 1[/itex].

Would anyone help to point out where I'm wrong or share some good references?

However, I came back to a point that confused me in the past and wanna get it clear now :)

One of the boundary conditions, regarding the magnetic fields parallel to the medium-interface is

[itex]H_{1//}-H_{2//}=J_s[/itex]

where [itex]H_{1//} \; and \; H_{2//} \; \text{are magnetic field strengths and} \; J_s \; \text{is the electric current per unit area} [/itex].

I found the derivation in my text book

*elements of engineering electromagnetics (6th edition)*by*Nannapaneni Narayana Rao*: (please view the attachment to find corresponding notations):[itex]\oint_{C} \vec{H} \cdot d\vec{l} = \int_{S} \vec{J} \cdot d\vec{s} + \frac{d \int_{S} \vec{D} \cdot d\vec{s}}{d \, t}[/itex]

Note that [itex]bc \approx 0 \; and \; da \approx 0[/itex] so that flux of electric fields is ignored, hence

[itex]ab \cdot H_{1//}+bc \cdot H_{\perp}+ cd \cdot H_{2//}+ da \cdot H_{\perp}= J_s \cdot AREA_{abcd}[/itex]

where [itex]ab=-cd, \; bc=-da[/itex], thus

[itex]ab \cdot H_{1//}- ab \cdot H_{2//} = J_s \cdot AREA_{abcd}[/itex]

and now, I just can't get [itex]H_{1//}-H_{2//}=J_s[/itex] because [itex]\frac{AREA_{abcd}}{ab} \not= 1[/itex].

Would anyone help to point out where I'm wrong or share some good references?