Electrostatic Boundary Conditions at the Interface of Two Medium

Electrostatic Boundary Conditions at the Interface of Two Medium

In this article, we will discuss the electrostatic boundary conditions at the interface of two medium. We will discuss the boundary conditions for the electric field \vec{E} as well as the displacement vector \vec{D}. At the heart of our discussion, we will use two fundamentally important understanding of the electrostatic field, namely electrostatic field is conservative force field and the differential form of Gauss’s divergence theorem.

We strongly recommend to check our article on Magnetostatic Boundary Conditions and start a comparative study.

As already mentioned that the electrostatic field is conservative in nature, therefore the Curl of the electric field should be zero.

(1)   \begin{equation*} \vec{\nabla }\times\vec{E}=0 \end{equation*}

and the integral form becomes

    \[\oint\vec{E}.\vec{dl}=0\]

We define the displacement vector as

    \[\vec{D}=\epsilon\vec{E}=\epsilon_0\epsilon_r\vec{E} \]

This description of the displacement vector as it includes permittivity of the medium is helpful while discussing the behavior of the electrostatic field within the medium. Here \epsilon_0 is the absolute permittivity of the free space and \epsilon_r is the relative permittivity of the corresponding medium.

Now we apply the integral form of the Curl theorem at the boundary of two dielectric mediums as shown in the diagram below

electrostatic boundary condition

 

    \[\vec{E_1}\cdot\vec{\Delta l}+\vec{E_1}\cdot\vec{ \frac{\Delta h}{2}}+\vec{E_2}\cdot\vec{\frac{\Delta h}{2}}+\vec{E_2}\cdot\vec{\Delta l}+\vec{E_2}\cdot\vec{\frac{\Delta h}{2}}+\vec{E_1}\cdot\vec{\frac{\Delta h}{2}}=0\]

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

    \[\vec{E_1}\cdot\vec{\Delta l}=\vec{E_2}\cdot\vec{\Delta l}\]

    \[E_{1t}=E_{2t}\]

where subscript t denotes the tangential component. Thus at the boundary of two medium, tangential component of the electric field is continuous.

    \[E_{1t}=E_{2t}\]

    \[\frac{D_{1t}}{\epsilon_{1r}}=\frac{D_{2t}}{\epsilon_{2r}}\]

Thus at the boundary, the tangential component of \vec{D} is discontinuous.

Tangential component of the electric field is continuous across the boundary however the displacement vector is discontinuous.

We will now use the integral form of the divergence theorem of the displacement vector at the boundary of two medium to figure out the fate of the normal component of fields.

electrostatic boundary condition

    \[\oint\vec{D}.\vec{da}=Q_{free}=\sigma_{free}A\]

    \[D_{1n}-D_{2n}=\sigma_{free}\]

Again at the boundary, we can reduce the height of the pillbox to be infinitesimally small. Here subscript n denotes the normal component of the field. If there is no free surface charge, we will have

    \[D_{1n}=D_{2n}\]

Thus the normal component of the displacement vector is continuous in absence of any free charge at the surface.

    \[\epsilon_{1r}E_{1n}=\epsilon_{2r}E_{2n}\]

Thus normal component of the electric vector is discontinuous at the boundary.

Normal component of the electric field is discontinuous across the boundary, however, the displacement vector is continuous in absence of any surface charge.

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