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    Magnetic Boundary Conditions

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    Suresh Thallapelli
    The document discusses magnetic boundary conditions at the interface between two materials with different magnetic permeabilities. The two key magnetic boundary conditions are: 1) The tangential component of the magnetic field is continuous across the boundary. 2) The normal component of the magnetic flux density is continuous across the boundary. If a surface current is present at the boundary, the tangential components of the magnetic field will not be continuous, but will instead be related by the boundary condition that the difference between the tangential fields is equal to the surface current.

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    Magnetic Boundary Conditions

    Uploaded by

    Suresh Thallapelli
    462 views6 pages
    The document discusses magnetic boundary conditions at the interface between two materials with different magnetic permeabilities. The two key magnetic boundary conditions are: 1) The tangential component of the magnetic field is continuous across the boundary. 2) The normal component of the magnetic flux density is continuous across the boundary. If a surface current is present at the boundary, the tangential components of the magnetic field will not be continuous, but will instead be related by the boundary condition that the difference between the tangential fields is equal to the surface current.

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    Boundary Conditions

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    The document discusses magnetic boundary conditions at the interface between two materials with different magnetic permeabilities. The two key magnetic boundary conditions are: 1) The tangential component of the magnetic field is continuous across the boundary. 2) The normal component of the magnetic flux density is continuous across the boundary. If a surface current is present at the boundary, the tangential components of the magnetic field will not be continuous, but will instead be related by the boundary condition that the difference between the tangential fields is equal to the surface current.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    462 views6 pages

    Magnetic Boundary Conditions

    Uploaded by

    Suresh Thallapelli
    The document discusses magnetic boundary conditions at the interface between two materials with different magnetic permeabilities. The two key magnetic boundary conditions are: 1) The tangential component of the magnetic field is continuous across the boundary. 2) The normal component of the magnetic flux density is continuous across the boundary. If a surface current is present at the boundary, the tangential components of the magnetic field will not be continuous, but will instead be related by the boundary condition that the difference between the tangential fields is equal to the surface current.

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    You are on page 1of 6

    11/28/2004 Magnetic Boundary Conditions 1/6

    Magnetic Boundary
    Conditions
    Consider the interface between two different materials with
    dissimilar permeabilities:

    H1 ( r ), B1 ( r )

    µ1

    H2 ( r ), B2 ( r )
    µ2

    Say that a magnetic field and a magnetic flux density is present


    in both regions.

    Q: How are the fields in dielectric region 1 (i.e.,


    H1 ( r ) , B1 ( r ) ) related to the fields in region 2 (i.e.,
    H2 ( r ) , B2 ( r ) )?

    A: They must satisfy the magnetic boundary


    conditions !

    Jim Stiles The Univ. of Kansas Dept. of EECS


    11/28/2004 Magnetic Boundary Conditions 2/6

    First, let’s write the fields at the interface in terms of their


    normal (e.g., Hn ( r ) ) and tangential (e.g., Ht ( r ) ) vector
    components:

    H1n ( r ) H1 ( r ) = H1t ( r ) + H1n ( r )

    ˆan
    µ1 H1t ( r )
    H2t ( r )
    H2n ( r )
    H2 ( r ) = H2t ( r ) + H2n ( r )
    µ2

    Our first boundary condition states that the tangential


    component of the magnetic field is continuous across a
    boundary. In other words:

    H1t ( rb ) = H2t ( rb )

    where rb denotes to any point along the interface (e.g., material


    boundary).

    Jim Stiles The Univ. of Kansas Dept. of EECS


    11/28/2004 Magnetic Boundary Conditions 3/6

    The tangential component of the magnetic field on


    one side of the material boundary is equal to the tangential
    component on the other side !

    We can likewise consider the magnetic flux densities on the


    material interface in terms of their normal and tangential
    components:

    B1n ( r ) B1 ( r ) = µ1 H1 ( r )

    ˆan
    µ1 B1t ( r )
    B2t ( r )
    B2n ( r )
    B2 ( r ) = µ2 H2 ( r )

    µ2

    The second magnetic boundary condition states that the normal


    vector component of the magnetic flux density is continuous
    across the material boundary. In other words:

    B1n ( rb ) = B2n ( rb )

    where rb denotes any point along the interface (i.e., the


    material boundary).

    Jim Stiles The Univ. of Kansas Dept. of EECS


    11/28/2004 Magnetic Boundary Conditions 4/6

    Since B ( r ) = µ H ( r ) , these boundary conditions can likewise be


    expressed as:

    H1t ( rb ) = H2t ( rb )

    B1t ( rb ) B2t ( rb )
    =
    µ1 µ2

    and as:

    B1n ( rb ) = B2n ( rb )

    µ1H1n ( rb ) = µ2H2n ( rb )

    Note again the perfect analogy to the boundary conditions of


    electrostatics!

    Jim Stiles The Univ. of Kansas Dept. of EECS


    11/28/2004 Magnetic Boundary Conditions 5/6

    Finally, recall that if a layer of free charge were lying at a


    dielectric boundary, the boundary condition for electric flux
    density was modified such that:

    aˆn ⋅ ⎡⎣D1 ( rb ) − D2 ( rb ) ⎤⎦ = ρs ( rb )
    D1n ( rb ) − D2n ( rb ) = ρs ( rb )

    There is an analogous problem in magnetostatics, wherein a


    surface current is flowing at the interface of two magnetic
    materials:

    ân
    H1t ( rb )
    µ1 Js ( rb )

    H2t ( rb )

    µ2

    In this case the tangential components of the magnetic field


    will not be continuous!

    Jim Stiles The Univ. of Kansas Dept. of EECS


    11/28/2004 Magnetic Boundary Conditions 6/6

    Instead, they are related by the boundary condition:

    ân x ( H1 ( rb ) − H2 ( rb ) ) = Js ( rb )

    This expression means that:

    1) H1t ( rb ) and H2t ( rb ) point in the same direction.

    2) H1t ( rb ) and H2t ( rb ) are orthogonal to Js ( rb ) .

    3) The difference between H1t ( rb ) and H2t ( rb ) is Js ( rb ) .

    Recall that H ( r ) and Js (r ) have the same units—


    Amperes/meter!

    Note for this case, the boundary condition for the magnetic
    flux density remains unchanged, i.e.:

    B1n ( rb ) = B2n ( rb )

    regardless of Js ( rb ) .

    Jim Stiles The Univ. of Kansas Dept. of EECS

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