Waiting
Login-Verarbeitung ...

Trial ends in Request Full Access Tell Your Colleague About Jove

29.16: Magnetostatic Boundary Conditions

TABLE OF
CONTENTS
JoVE Core
Physics

Ein Abonnement für JoVE ist erforderlich, um diesen Inhalt ansehen zu können. Melden Sie sich an oder starten Sie Ihre kostenlose Testversion.

Education
Magnetostatic Boundary Conditions
 
TRANSCRIPT

29.16: Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous across any boundary; hence the derivative of the vector potential is discontinuous across the boundary.

Consider two linear magnetic media with permeability μ1 and μ2, separated by an interface. The magnetic field intensity in medium 1 at a point P1 has a magnitude H1, and makes an angle α1 with the normal. Similarly, the magnetic field intensity in medium 2 at a point P2 has a magnitude H2, and makes an angle α2 with the normal.

Figure1

The zero divergence of the magnetic field ensures that the normal component of the magnetic field is continuous across the boundary. The magnetic field is proportional to the magnetic field intensity for linear magnetic media. Thus, the boundary conditions for the normal component of the magnetic field intensity are obtained.

Rewriting the normal components of the magnetic field in terms of the angles α1 and α2,

Equation1

Since both media possess finite conductivities, free surface currents do not exist at the interface. Thus, the tangential component of the magnetic field intensity is continuous across the boundary. Again, the tangential components can be rewritten in terms of the angles α1 and α2:

Equation2

Taking the ratio of normal and tangential components, a new expression is derived:

Equation3

This expression is similar to the expression obtained for boundary conditions for two dielectric interfaces, except that the permittivity replaces the permeability.

If the first medium is a non-magnetic medium, like air, and the second one is a magnetic medium, the permeability of the second medium is greater than the permeability of the first medium. Thus, the angle α2 approaches ninety degrees. This implies that, for any arbitrary angle α1 that is not close to zero, the magnetic field in a ferromagnetic medium runs parallel to the interface. If the first medium is ferromagnetic and the second is non-magnetic, α2 approaches zero. This implies that if a magnetic field originates in a ferromagnetic medium, the magnetic flux lines emerge into the air medium in a direction almost normal to the interface.


Suggested Reading

Tags

Keywords: Magnetic Field Magnetic Medium Boundary Conditions Normal Component Tangential Component Permeability Surface Current Vector Potential Magnetic Field Intensity

Get cutting-edge science videos from JoVE sent straight to your inbox every month.

Waiting X
Simple Hit Counter