Substitute the boundary conditions part part

Substitute Go 6983 boundary conditions ∂Φ∂y y=+0−∂Φ∂y y=−0=σ and ∂Φ∂y y=−d+0−∂Φ∂y y=−d−0=−σ into Eq. (2) and let α = 2π/λ getsequation(3)Φ1=σs4πλsin(αx)e−αy  y>0Φ2=σs4πλsin(αx)[1+eαdeαd−e−αdeαy−1+e−αdeαd−e−αde-αy] -d < y<0Φ3=−σs4πλsin(αx)eαdeαy  y<−d
So the demagnetizing field inside the film isH?d=−∇Φ=−(∂Φ∂xiˆ+∂Φ∂yjˆ)equation(4)=−σs2cos(αx)(1+eαdeαd−e−αdeαy−1+e−αdeαd−e−αde-αy)iˆ−σs2sin(αx)(1+eαdeαd−e−αdeαy+1+e−αdeαd−e−αde-αy)jˆ  for−d<y<0
5. The magnetization distribution of stripe domains film
According to the hysteresis loop of Fig. 3(a) and the MFM images of Fig. 4, the magnetization process of the stripe domains is the typical magnetization rotation process. According to the coordinate system shows as Fig. 7(a), the magnetization vectors M can be expressed asequation(5)Mx=MssinθcosφMy=MssinθsinφMz=Mscosθwhere θ and φ are the function of the position (x, y, z). In saturation state, the magnetization vector M is along the external magnetic field direction z axis (along the stripe direction). With the decrease of the external magnetic field, because of the effect of the demagnetizing field, the M deviates from z axis. According to the boundary condition of the static magnetic field, the magnetic charges of the film surface areequation(6)σ(x,z)=n?⋅M?=My