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Physics LibreTexts

10.2: Boundary Conditions

  • Page ID
    22721
  • 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 Et1 is the field component parallel with L in material number 1 (vacuum in this case) and Et2 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 10.3.PNG
    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 10.4.PNG
    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:

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