Magnetostatic Boundary Conditions at the Interface of Two Medium

Magnetostatic Boundary Conditions at the Interface of Two Medium

In this article, we will discuss the magnetostatic boundary conditions at the interface of two medium. However we will start from two fundamentally important concepts of the magnetostatic field, we strongly recommend to check our article on Electrostatic boundary conditions and make a comparative study of the two. As a consequence of the fact that magnetic monopole does not exist, we have zero divergences corresponding to any magnetic field. Moreover, the magnetic field is not conservative and we have a nonvanishing curl. We will use these two conditions to figure out magnetostatic boundary conditions.

As already mentioned that the magnetostatic field is non-conservative in nature, therefore the Curl of the magnetic field should not vanish and we have

(1)   \begin{equation*} \vec{\nabla }\times\vec{H}= \vec{J}_f \end{equation*}

where \vec{J}_f is free surface current density. In the integral form it becomes,

    \[\oint\vec{H}.\vec{dl}=I_f\]

where I_f is free current enclosed by the loop. Now we consider the interface of two medium as shown in the figure below and apply the integral form of the curl theorem. The top half of the loop is in the first medium whereas the bottom half of the loop is in the second medium.

magnetostatic boundary condition

    \[\vec{H_1}\cdot\vec{\Delta l}+\vec{H_1}\cdot\vec{ \frac{\Delta h}{2}}+\vec{H_2}\cdot\vec{\frac{\Delta h}{2}}+\vec{H_2}\cdot\vec{\Delta l}+\vec{H_2}\cdot\vec{\frac{\Delta h}{2}}+\vec{H_1}\cdot\vec{\frac{\Delta h}{2}}=I_f\]

To calculate the boundary condition we make the height of the loop to be infinitesimally small or \Delta h \to 0,

    \[\vec{H_1}\cdot\vec{\Delta l}+\vec{H_2}\cdot\vec{\Delta l}=I_f\]

    \[H_{1t}-H_{2t}=J_f\]

where subscript t denotes the tangential component. Thus at the interface of two medium, tangential component of the magnetic field is discontinuous and in absence of any free surface current, the tangential component of the magnetic field is continuous.

    \[H_{1t}=H_{2t}\]

Tangential component of the external magnetic field is discontinuous at the interface of two medium.

Magnetic monopole does not exist and consequently the divergence of the magnetic field vanishes. We will use the integral form of this result to find out the behavior of the normal component of the magnetic field at the interface of two medium.

magnetostatic boundary condition

    \[\oint\vec{B}.\vec{da}=0\]

    \[\vec{B}_1\cdot\vec{A}+\vec{B}_2\cdot\vec{A}=0\]

    \[B_{1n}-B_{2n}=0\]

Here subscript n denotes the normal component. Again we have reduced the height of the pillbox to be infinitesimally small and correspondingly the area of the curved surface vanishes.

Thus, the normal component of the magnetic field is continuous at the boundary.

Normal component of the magnetic field is always continuous at the interface of two medium.

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