In the tutorial Fields at boundaries, parallel and perpendicular boundary condit
ID: 3163485 • Letter: I
Question
In the tutorial Fields at boundaries, parallel and perpendicular boundary conditions were derived using a flat boundary between two regions, each with a different uniform field. However, surfaces arc typically curved and between two regions of non-uniform field. Explain in your own words why the boundary conditions arc still true for the curved boundary between non-uniform fields. (I.e., why can you approximate the general case as a flat boundary between two regions of uniform field when considering boundary conditions?) Consider the vector derivatives for the static displacement field D and the static auxiliary field H: nabla middot D = rho_free nabla times D = nabla times P nabla middot H = - nabla middot M nabla times H = J_free a. Boundaries are often considered across the surface of a magnetizable or polarizable material. Which of the above vector derivatives are zero at this type of surface? Explain. b. Based on your answer above, which of the following must be Perpendicular continuous across a boundary: the perpendicular component of H, and or the parallel component of H?Explanation / Answer
Ans:- (1). the parallel & perpendicular boundary condition are flat between two region with different field. so our surface are curve between two region for non uniform field. the boundary condition curved boundary for non uniform fields. The concept of boundary condition for curve surface in non uniform field for different from the approach take for the different types of surface boundary conditions for parallel & perpendicular boundary condition can only be applied on reverse side, non uniform boundaries of an atomic block as well as multi-block region. This nature of surface boundary is curve. so works also only if multi-block has applied in non uniform field. so for that case, all curve surface between two region for non uniform filed which are touch with outside non boundary of the multi-block can be curve surface, if the boundary corresponding region on the reverse side wall is non uniform region. In that condition an different -block . Surface curve is normally are property of the stream for different operator. It has lot of effect for parallel & perpendicular region. In this case of a multi-block, curve surface can be affect parallel & perpendicular region. Being non uniform field that if you define a non- curve region which located nearest neighbor boundary, the value of that boundary neighbor for curve surface is calculated by non boundary region along with outer boundary.
(2). the magnetizable or polarizable material condition. the vector derivative are zero so we use Usadel equation
D (g g) + i [ 3 + g ] =0 so that D =diffusion coefficient
dot matrix vector condition = vector V . vector D = fm
cross matrix vector condition = vector V X vector D = vector V X vector P
while all matrix for given condition explain the above vector dot matrix vector & cross matrix vector are zero. In general condition it can solved by functional of the different color function matrix , i.e. . In the special case of a magnetizable or polarizable material condition .so we describes the magnetizable or polarizable material and direction of the magnetic field in different type of vector Pauli matrices. In the present of accuracy gauge fields such as magnetizable or polarizable material vector potential A vector so our magnatizable magnetic field for can be change the gradient vector. so we prove that the magnetizable or polarizable material all kind of matrix dot & cross are to be zero for surface.
(3). the continuous cross the boundary perpendicular component D & H also parallel component D & H. The D & H perpendicular boundary when a pair of lines as sown in image normally they are assumes that all plane lie in same boundary. so that reasonable for perpendicular vector D & H are coplanar lines are perpendicular and parallel ever been used. suppose we are assume a box. The lines AB and CG don’t lie in similar region, so they are dot & cross skew lines, so that they seem to be perpendicular & parallel in some boundary condition. so that the show in arbitrary dimensions by explain in the linear algebra. Let D is the displacement vector from A B. H is the displacement vector from CG . so this explanation angle are also between these vectors is 90 . Continuous across the boundary perpendicular & Parallel in requires the coplanar region.