10.2: Boundary Conditions

10.2.1 The Tangential Components of the Electric Field.

Apply Stokes’ theorem to the Maxwell equation

$$\operatorname{curl}(\vec{\text{E}})=-\frac{\partial \vec{\text{B}}}{\partial \text{t}} \nonumber$$

and the small loop whose sides are L long and $\delta$ long as shown in Figure (10.1.2):

$$\oint \vec{\text{E}} \cdot \vec{\text{d} \text{L}}=-\frac{\partial}{\partial \text{t}} \int \int_{\text {Area }} \vec{\text{B}} \cdot \text{d} \vec{\text{A}}. \nonumber$$

One then takes the limit as the sides $\delta$ shrink to zero. The line integral of the electric field gives

$$\oint \vec{\text{E}} \cdot \text{d} \vec{\text{L}}=\left(\text{E}_{\text{t} 1}-\text{E}_{\text{t} 2}\right) \text{L}, \nonumber$$

where E_t1 is the field component parallel with L in material number 1 (vacuum in this case) and E_t2 is the electric field component parallel with L in material number 2. The flux of the magnetic field through the loop goes to zero as $\delta$ goes to zero, therefore

$$\left(\text{E}_{\text{t} 1}-\text{E}_{\text{t} 2}\right)=0 \nonumber$$

Figure $\PageIndex{3}$: The Maxwell equation $\operatorname{curl}(\vec{\text{H}})=-\partial \vec{\text{D}} / \partial \text{t}$ requires the tangential components of $\vec H$ to be continuous across any interface. See the text.

Figure $\PageIndex{4}$: The Maxwell equation $\operatorname{div}(\vec{\text{B}})=0$ requires the normal component of $\vec B$ to be continuous across any interface. See the text.

or

$$\text{E}_{\text{t} 1}=\text{E}_{\text{t} 2} . \label{10.23}$$

At the boundary between two materials the transverse components of $\vec E$ must be continuous.

10.2.2 The Tangential Components of the Magnetic Field.

Apply Stokes’ theorem to a small loop as shown in fig(10.2.3):

$$\operatorname{curl}(\vec{\text{H}})=\frac{\partial \vec{\text{D}}}{\partial \text{t}} , \nonumber$$

where it has been assumed that there are no free currents in either material, and no surface free current density on the interface between material number(1) and material number(2). Therefore

$$\oint_{C} \vec{\text{H}} \cdot \text{d} \vec{\text{L}}=\frac{\partial}{\partial \text{t}} \int \int_{A r e a} \vec{\text{D}} \cdot \vec{\text{d} \text{S}} . \nonumber$$

Upon taking the limit as $\delta$ shrinks to zero the surface integral over $\vec D$ gives nothing and

$$\left(\text{H}_{\text{t} 1}-\text{H}_{\text{t} 2}\right) \text{L}=0 , \nonumber$$

that is

$$\text{H}_{\text{t} 1}=\text{H}_{\text{t} 2} . \label{10.24}$$

The transverse components of the magnetic field $\vec H$ must be continuous across the boundary between two materials.

10.2.3 The Normal Component of the Field B.

The normal component of the magnetic field $\vec B$ must be continuous across any interface as a consequence of the Maxwell equation $d i v(\vec{\text{B}})=0$; see Figure (10.2.4). In Figure (10.2.4) Gauss’ theorem is applied to a small pill-box that spans an arbitrary surface. The height of the pill-box, $\delta$, is taken to be so small that any contributions to the surface integral from the sides of the box can be neglected. The continuity of the normal component of $\vec B$ is then forced by the requirement that the surface integral of $\vec B$ over the pill-box be zero:

$$\text{B}_{\text{n} 1}=\text{B}_{\text{n} 2} . \label{10.25}$$