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 co…
Inside a homogenous magnetic medium, the magnetic field is continuous. However, on crossing a boundary, it becomes discontinuous.
Consider a Gaussian pillbox across the boundary. The zero divergence of the magnetic field ensures that the total magnetic flux enclosed within the pillbox is zero. So, the normal components of the field are equal across the boundary.
Now, consider an Amperian loop perpendicular to the surface current and apply Ampere's Law to each loop section.
The line integral of the magnetic field exists only for the loop's top and bottom sections since the loop width tends to zero. So, the field's tangential component is discontinuous.
The discontinuity of the tangential components can be expressed in vectorial form.
Now, consider an interface between two media. The divergence of magnetic field intensity equals the negative divergence of the magnetization. So, the normal component of magnetic field intensity is discontinuous.
Consider an Amperian loop perpendicular to the surface current. Applying Ampere's law, the tangential component of magnetic field intensity is discontinuous. This discontinuity can be expressed in vector form.
View the full transcript and gain access to JoVE Core videos
Q1: Why is the normal component of magnetic field continuous across a boundary between two media?
The zero divergence of the magnetic field ensures that the total magnetic flux entering and leaving a Gaussian pillbox at the boundary must be equal. This means the normal components of the magnetic field on both sides of the boundary are identical, making the normal component continuous regardless of the media properties.
Q2: What causes the tangential component of magnetic field to be discontinuous at a boundary?
Surface currents at the boundary create a discontinuity in the tangential component of the magnetic field. Using an Amperian loop perpendicular to the surface current, Ampere's Law shows that the tangential field component jumps by an amount equal to the product of vacuum permeability and the surface current density.
Q3: How do boundary conditions differ for magnetic field intensity versus magnetic field in two media?
For magnetic field intensity, the normal component is discontinuous when magnetization differs between media, while the tangential component remains continuous if no free surface currents exist. In contrast, for magnetic field itself, the normal component is continuous due to zero divergence, but the tangential component is discontinuous at surface currents.
Q4: What happens to magnetic field lines at an interface between a ferromagnetic medium and air?
Magnetic field lines emerging from a ferromagnetic medium into air approach a direction nearly normal to the interface. This occurs because the permeability ratio causes the refraction angle to approach zero, forcing magnetic flux lines to exit perpendicular to the boundary rather than at oblique angles.
Q5: How does the magnetic vector potential behave across a boundary between two media?
The magnetic vector potential is continuous across any boundary between two media, similar to the scalar potential in electrostatics. However, the derivative of the vector potential is discontinuous at the boundary, reflecting the discontinuity in the tangential component of the magnetic field.
Q6: What is the relationship between permeability and magnetic field refraction at a boundary?
The ratio of tangential to normal components of magnetic field intensity is proportional to the permeability of each medium. This relationship mirrors Snell's Law for dielectrics, where higher permeability in one medium causes the magnetic field to refract more parallel to the interface, similar to how refractive index affects light bending.
Q7: Why is the normal component of magnetic field intensity discontinuous at an interface between two magnetic media?
The divergence of magnetic field intensity equals the negative divergence of magnetization at the interface. When magnetization differs between the two media, this creates a discontinuity in the normal component of magnetic field intensity, even though the normal component of the magnetic field itself remains continuous.