S.6 The plane boundary defined by z = 0 separates air from a block of iron. IfB = 4x – 6? + 82 in ...

Expert Answer

We are given a magnetic field B=4x−6y+8z\mathbf{B} = 4x - 6y + 8zB=4x−6y+8z in the air (for z>0z > 0z>0) and the magnetic permeability μ=5000 H/m\mu = 5000 \, \text{H/m}μ=5000H/m in iron (for z<0z < 0z<0), with the condition that no surface current exists on the boundary at z=0z = 0z=0.

We need to find the magnetic field B\mathbf{B}B in the iron region (z<0z < 0z<0).

Key Concepts

1. Boundary conditions for magnetic fields:
   • Continuity of the tangential component of the magnetic field: The tangential components of the magnetic field B\mathbf{B}B must be continuous across the boundary.
   • Normal component of the magnetic flux density B\mathbf{B}B: The normal component of B\mathbf{B}B can change across the boundary, but it must satisfy the equation n^⋅(B2−B1)=0\hat{n} \cdot (\mathbf{B}_2 - \mathbf{B}_1) = 0n^⋅(B2?−B1?)=0, where n^\hat{n}n^ is the unit vector normal to the boundary, and B1\mathbf{B}_1B1? and B2\mathbf{B}_2B2? are the magnetic flux densities on either side of the boundary.