Tangential component: In order to derive boundary condition at current interface, we will apply ‘Ampere’s law’

$∮\bar{H}.d \bar{l}=I_{enclosed}=0 \\ ∫\bar{H}_{t1}.∆ \bar{w}+ ∫\bar{H}_{t2}.∆ \bar{L}+ \bar{H}_{t2}.∆ \bar{w}- \bar{H}_{t1}.∆ \bar{L}=0 \\ \bar{H}_{t1}.∆ \bar{w}=\bar{H}_{t2}.∆ \bar{w}=0 \\ ∫H_{t1}.∆ L= ∫H_{t2}.∆ L \\ \bar{H}_{t1}= \bar{H}_{t2}$

Normal components:

For normal components use $∮\bar{B} .d \bar{s}= ∅$

$-∮B_{n1}.d \bar{s}+∮B_{n1}.d \bar{s}+ ∮B_{n2}.d \bar{s}= ∅ \\ ∮B_{n1}.d \bar{s}= ∮B_{n2}.d \bar{s} ….[θ=90^o-∮B_{n1}.d \bar{s}=0 ,I=0- ∅=0 ] \\ B_{n1}= B_{n2}$

The normal component of flux density at current free interface is continuous