2.7: Jump Conditions
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Systems having nonuniform properties are often modeled by regions of uniform properties, separate by boundaries across which these properties change abruptly. Fields are similarly often given a piecewise representation with jump conditions used to "splice" them together at the discontinuities. These conditions, derived here for reference, are implied by the integral laws. They guarantee that the associated differential laws are satisfied through the singular region of the discontinuity.
Electroquasistatic Jump Conditions
A section of the boundary can be enclosed by a volume element having the thickness \(\Delta\) and cross-sectional area \(A\), as depicted by Figure 2.10.1. The linear dimensions of the cross-sectional area \(A\) are, by definition, much greater than the thickness \(\Delta\). Implicit to this statement is the assumption that, although the surface can be curvilinear, its radius of curvature must be much greater than a characteristic thickness over which variations in the properties and fields take place.
The normal vector \(\overrightarrow{n}\) used in this section is a unit vector perpendicular to the boundary and direct from region b to region a, as.shown in Figure 2.10.1. Since this same symbol is used in connection with integral theorems and laws to denote a normal vector to surfaces of integration, these latter vectors are denoted by \(\overrightarrow{1_n}\).
First, consider the boundary conditions implied by Gauss' law, Equation 2.3.23a, with Equation 2.8.3 used to introduce \(\rho_p\). This law is first multiplied by \(\nu^m\) and then integrated over the volume \(V\):
\[ \int_{V} \nu^m \nabla \cdot \varepsilon_o \overrightarrow{E} dV = \int_{V} \nu^m \rho_f dV + \int_{V} \nu^m \rho_p dV \label{1} \]
Here, \(v\) is a coordinate (like \(x\),\(y\), or \(z\)) perpendicular to the boundary and hence in the direction of \(\overrightarrow{n}\), as shown in Figure 2.10.1.
First, consider the particular case of Equation \ref{1} with \(m = 0\). Then, the integration gives
\[ \overrightarrow{n} \cdot \big[\big] \varepsilon_o E \big[\big] = \sigma_f + \sigma_p \label{2} \]
where \(\big[\big] \overrightarrow{A} \big[\big] \equiv \overrightarrow{A^a} - \overrightarrow{A^b} \) and \(\big[\big]\psi\big[\big] \equiv \psi^a - \psi^b\) and the free surface charge density \(\sigma_f\) and polarization surface charge density \(\sigma_p\) have been defined as
\[ \sigma_f = \lim_{A \to 0} \frac{1}{A} \int \rho_f dV, \quad \sigma_p = \lim_{A \to 0} \frac{1}{A} \int \rho_p dV \label{3} \]
The relationship between the surface charge and the electric field intensity normal to the boundary can be pictured as shown in Figure 2.10.2b.
In view of Equation \ref{2}, the normal electric field intensity is continuous at the interface unless there is a singularity in charge. Thus, with volume charges to either side of the interface, there is an abrupt change in the rate of change of the electric field intensity normal to the boundary, but the field is itself continuous. On the other hand, as illustrated by the sketches of Figure 2.10.2b, if there is an appreciable charge per unit area within the boundary, the electric field intensity is discontinuous, and undergoes a step discontinuity.
A somewhat less familiar situation is that of Figure 2.10.2c. Within the boundary there are regions of large positive and negative charge concentrations with an associated intense electric field between. In the limit where the boundary becomes very thin, a component of the surface charge density becomes a doublet, and the electric field becomes an impulse.
The double layer can be pictured as being positive surface charges disposed on one side of the boundary, and negative surface charges distributed on the other, with an internal component of the electric field originating on the positive charges and terminating on the negative ones. The magnitude of the double layer is equal to the product of the positive surface charge density and the distance between these layers, \(\Delta\). In the limit where the layer thickness becomes infinitely thin while the double-layer magnitude remains constant, the electric field within the double layer must approach infinity. Thus, associated with the doublet of charge density, there is an impulse in the electric field intensity, as sketched in Figure 2.10.2c.
The boundary condition to be used in connection with a double layer is found from Equation \ref{1} by letting \(m = 1\). The left-hand side of Equation \ref{1} can be integrated by parts, so that it becomes
\[ \int_{V} \nabla \cdot (\varepsilon_o \nu \overrightarrow{E}) dV - \int_{V} \varepsilon_o \overrightarrow{E} \cdot \nabla \nu dV = \int_{V} \nu (\rho_f + \rho_p) dV \label{4} \]
For the incremental volume, the surface double layer density is defined as
\[ P_{\Sigma} = \lim_{A \to 0} \frac{1}{A} \int \nu (\rho_f + \rho_p) dV = \int_{\nu_-}^{\nu_+} \nu (\rho_f + \rho_p) d\nu \label{5} \]
and so the right-hand side of Equation \ref{4} is \(Ap_{\Sigma}\). The origin of the \(\Delta\) axis remains to be defined but \(\Delta \equiv \nu_+ - \nu\). To glean a jump condition from the equation, the second EQS law is incorporated. That \(\overrightarrow{E}\) is irrotational Equation 2.3.24a, is represented by defining the electric potential
\[ \overrightarrow{E} = -\nabla \phi \label{6} \]
Thus, the second term on the left in Equation \ref{4} becomes
\[ \int_{V} \varepsilon_o E \cdot \nabla \nu dV = - \int_{V} \varepsilon_o \nabla \phi \cdot \nabla \nu dV = - \int_{V} \varepsilon_o \nabla \cdot (\phi \nabla \nu) dV + \int_{V} \varepsilon_o \phi \nabla^2 \nu dV \label{7} \]
Evaluation of \(\nabla^2 \nu\) gives nothing because v is defined as a local Cartesian coordinate. The last integral vanishes, and with the application of Gauss' theorem, Equation 2.6.2, it follows that Equation \ref{4} becomes
\[ \oint_{S} \varepsilon_o \nu \overrightarrow{E} \cdot \overrightarrow{1_n} da + \oint_{S} \varepsilon_o \phi \nabla \nu \cdot \overrightarrow{1_n} da = Ap_{\Sigma} \label{8} \]
Provided that within the layer, \(\overrightarrow{E}\) parallel to the interface and \(\phi\) are finite (not impulses in the limit \(\Delta \rightarrow 0\)), Equation \ref{8} only has contributions to the surface integrals from the regions to either side of the interface. Thus,
\[ A\varepsilon_o (\nu_+ \overrightarrow{E^a} - \nu_- \overrightarrow{E^b}) \cdot \overrightarrow{n} + A \varepsilon_o \big[\big] \phi \big[\big] = Ap_{\Sigma} \label{9} \]
The origin of the \(\nu\) axis is adjusted to make the first term vanish. The required boundary condition to be associated with Eqs. 2.3.23a and 2.3.23b is
\[ \varepsilon_o \big[\big] \phi \big[\big] = p_{\Sigma} \label{10} \]
The gradient of Equation \ref{10} within the plane of the interface converts the jump condition to one in terms of the electric field:
\[ \varepsilon_o \big[\big] \overrightarrow{E_t} \big[\big] = - \nabla_{\Sigma} p_{\Sigma} \label{11} \]
Here \(\nabla_{\Sigma}\) is the surface gradient and t denotes components tangential to the interfacial plane.
In the absence of a double-layer surface density, these last two boundary conditions are the familiar statement that the tangential electric field intensity at a boundary must be continuous. The statement given in Equation \ref{10} that the potential must be continuous at a boundary is another way of stating this requirement on the tangential electric field intensity. With a double layer, the tangential electric field intensity is discontinuous, as is also the potential.
Equations \ref{10} and \ref{11} could also be derived using the condition that the line integral of the electric field intensity around a closed loop intersecting the boundary vanish. Usually, the tangential electric field is continuous because there is no contribution to this line integral from those segments of the contour passing through the boundary. However, with the double layer, the electric field intensity within the boundary is infinite; so, even though the segments of the line integral across the boundary vanish as \(\Delta \rightarrow 0\), there is a net contribution from these segments of the integration.
It is clear that higher order singularities could also be handled by considering values of \(m\) in Equation \ref{1} greater than unity. However, the doublet is as singular a charge distribution as of interest physically.
There are two reasons for wishing to include the doublet charge distribution, one mathematical and one physical. Just as the surface charge density is a singularity in the volume charge density which can be used to terminate a normal electric field intensity at a boundary, the double layer is a termination of a tangential electric field. On the physical side, there are many situations in which a double layer actually exists within a very thin region of material. Double layers abound at interfaces between liquids and metals and between metals. The double-layer concept is useful for modeling electromechanical coupling involving these interfacial regions.
So far, those EQS laws have been considered that do not explicitly involve time rates of change. Conservation of charge does involve a dynamic term. Its associated boundary conditions can therefore be derived only by making further stipulations as to the nature of the boundary. It is now admitted that the boundary can, in general, be one which is deforming. Because time did not appear explicitly in the previous derivations of this section, the conditions derived are automatically appropriate, even if the boundary is moving.
The integral form of charge conservation, Equation 2.7.3a, is written for a volume \(V\) and surface \(S\) tied to the material itself. Thus, with \(\overrightarrow{v_s} \rightarrow \overrightarrow{v}\)
\[ \oint_{S} (\overrightarrow{J_f} - \rho_f \overrightarrow{v}) \cdot \overrightarrow{i_n} da = -\frac{d}{dt} \int_{V} \rho_f dV \label{12} \]
As seen in Figure 2.10.1, the volume of integration always encloses material of fixed identity and intersects the boundary. Implicit to this statement is the assumption that the boundary is one of demarcation between material regions. The material velocity is presumed to at most have a step singularity across the boundary. (It is important to recognize that there are other types of boundaries. For example, the boundary could be a shock front, with a gas moving through from one side of the interface to the other. In that case, the boundary conditions thus far derived would remain correct, because no mention has yet been made of the physical nature of the boundary.)
The left-hand side of Equation \ref{12} can be handled in a manner similar to that already illustrated, since it does not involve time rates of change. The integration is divided into two parts: one over the upper and lower surfaces of the volume, the other over the parts of the surface which intersect the boundary. The contributions to a current flow through these side surfaces comes from a surface current. It follows by using a two-dimensional form of Gauss' theorem, Equation 2.6.2, that the left-hand side of Equation \ref{12} is
\[ \int_{S^{'} + S^{''}} (\overrightarrow{J_f} - \rho_f \overrightarrow{v}) \cdot \overrightarrow{i_n} da + \int_{S^{'''}} (\overrightarrow{J_f} - \rho_f \overrightarrow{v}) \cdot \overrightarrow{i_n} da = A { \overrightarrow{n} \cdot \big[\big] \overrightarrow{J_f} - \overrightarrow{v}\rho_f \big[\big] + \nabla_{\Sigma} \cdot (\overrightarrow{K_f} - \sigma_f \overrightarrow{v_t})} \label{13} \]
Here, \(A\) is the area of intersection between the volume element and the boundary. The right-hand side of Equation \ref{12} is, by the definition of Equation \ref{3},
\[ \frac{d}{dt} \int_{V} \rho_f dV = \frac{d}{dt} \int_{A} \sigma_f da \label{14} \]
Note that, if the volume of integration \(V\), and hence the area of integration \(A\), is one always fixed to the material, then the area A is time-varying. The surface charge density is a function only of the two dimensions within the plane of the interface. Thus, the term on the right in Equation \ref{14} is a time derivative of a two-dimensional integral. This is a two-dimensional special case of the situation described by the generalized Leibnitz rule, Equation 2.6.5, which stated how the time derivative of a volume integral could be represented, even if the volume of integration were time-varying. Thus, Equation \ref{14} becomes
\[ \frac{d}{dt} \int_{V} \rho_f dV = A \Big[ \frac{\partial{\sigma_f}}{\partial{t}} + \nabla_{\Sigma} \cdot (\overrightarrow{v_t} \sigma_f) \Big] \label{15} \]
Finally, with the use of Eqs. \ref{13} and \ref{15}, Equation \ref{12} becomes the required jump condition representing charge conservation:
\[ \overrightarrow{n} \cdot \big[\big] \overrightarrow{J_f} - \rho_f \overrightarrow{v} \big[\big] + \nabla_{\Sigma} \cdot \overrightarrow{K_f} = - \frac{\partial{\sigma_f}}{\partial{t}} \label{16} \]
By contrast with Eqs. \ref{10} and \ref{11}, the expression is specialized to interfaces that do not support charge distributions so singular as a double layer. In using Equation \ref{16}, note that a partial derivative with respect to time is usually defined as one taken holding the spatial coordinates constant. A review of the derivation of Equation \ref{16} will make it clear that such is not the significance of the partial derivative on the right in Equation \ref{16}. The surface charge density is not defined throughout the three-dimensional space. Thus, this derivative means the partial derivative with respect to time, holding the coordinate within the plane of the interface constant.
The component of current normal to the boundary represented by the first term in Equation \ref{16} will be recognized as the free current density in a frame of reference moving with the boundary. A good question would be, "why is it that the normal current density appears in Equation \ref{16} evaluated in the primed frame of reference, while the surface free current density is not?" The answer points to the physical situation for which Equation 16 is appropriate. As the material boundary moves in the normal direction, the material ahead and behind carries a charge distribution along, but one that never reaches the boundary. By contrast, materials can flow in and out within the surface of the volume of interest, and carry with them a surface charge density of a convective nature. Thus, the surface divergence appearing in the second term of Equation \ref{16} can include both a conduction surface current and a convection surface current.
Magnetoquasistatic Jump Conditions
The integral forms of Ampere's law and Gauss' law for magnetic fields incorporate no time rates of change. Hence, the jump conditions implied by these laws are familiar from elementary electrodynamics. Ampere's law, Equation 2.7.1b, is integrated over the surface \(S\) and around the contour \(C\) enclosing the boundary, as sketched in Fig, 2.10.3, to obtain
\[ \overrightarrow{n} \times \big[\big] \overrightarrow{H} \big[\big] = \overrightarrow{K_f} \label{17} \]
where \(\overrightarrow{K_f}\) is the surface current density. Although it is entirely possible to consider a doublet of current density as a model, this impulsive singularity in the distribution of free current density is of as high an order as necessary to model MQS electromechanical situations of general interest.
From Gauss' law for magnetic fields, Equation 2.7.2b, applied to the incremental volume enclosing the interface, Figure 2.10.1, the jump condition is
\[ \overrightarrow{n} \times \big[\big] \mu_o (\overrightarrow{H} + \overrightarrow{M} ) \big[\big] = 0 \label{18} \]
Faraday's law of induction brings into play the time rate of change, and it is expected that motion of the boundary leads to an addition to the jump condition not found for stationary media. According to Equation 2.7.3b, the integral form of Faraday's law, for a contour fixed to the material (of fixed identity) so that \(\overrightarrow{v_s} \rightarrow \overrightarrow{v}\), is
\[ \oint_{C} (\overrightarrow{E} + \overrightarrow{v} \times \mu_o \overrightarrow{H}) \cdot \overrightarrow{d} l = -\frac{d}{dt} \int_{S} \mu_o (\overrightarrow{H} + \overrightarrow{M}) \cdot \overrightarrow{n} da \label{19} \]
With Equation \ref{19}, it has already been assumed that the boundary is a material one. Consistent with Equation \ref{17} is the assumption that it can be carrying a surface current with it as it deforms. If the surface \(S\) were not one of fixed identity, this would mean that the surface integral on the right could be a step function of time as the boundary passed through the surface of integration. The result would be a temporal impulse on the right which would make a contribution to the boundary condition even in the limit where the surface \(S\) becomes vanishingly small. By contrast, because the surface \(S\) is one of fixed identity, in the limit where the surface area vanishes, the right-hand side of Equation \ref{19} makes no contribution.
With the assumption that fields and velocity are at most step functions across the boundary, the integral on the left in Equation \ref{19} gives
\[ \overrightarrow{n} \times \big[\big] \overrightarrow{E} + \overrightarrow{v} \times \mu_o \overrightarrow{H} \big[\big] = 0 \label{20} \]
This expression is what would be expected, in view of the transformation law for the electric field in the MQS system. It states that \(\overrightarrow{E_t^{'}}\) is continuous across the interface.
Summary of Electroquasistatic and Magnetoquasistatic Conditions
Table 2.10.1 summarizes the jump conditions.
| EQS | MQS |
|---|---|
|
\[ \begin{align} &\overrightarrow{n} \cdot \big[\big] \varepsilon_o \overrightarrow{E} + \overrightarrow{P} \big[\big] = \sigma_f \nonumber \\ &\overrightarrow{n} \cdot \big[\big] \overrightarrow{P} \big[\big] = -\sigma_p \nonumber \end{align} \nonumber \] |
\[ \overrightarrow{n} \times \big[\big] \overrightarrow{H} \big[\big] = -K_f \label{21} \] |
|
\[ \begin{align} &\varepsilon_o \cdot \big[\big] \phi \big[\big] = \sigma_d \nonumber \\ &\varepsilon_o \cdot \big[\big] \overrightarrow{E_t} \big[\big] = -\nabla_{\Sigma} \sigma_d \nonumber \end{align} \nonumber \] |
\[ \begin{align} &\overrightarrow{n} \cdot \mu_o \big[\big] \overrightarrow{H} + \overrightarrow{M} \big[\big] = 0 \nonumber \\ &\overrightarrow{n} \cdot \mu_o \big[\big] \overrightarrow{M} \big[\big] = -\sigma_m \nonumber \end{align} \label{22} \] |
| \[ \overrightarrow{n} \cdot \big[\big] \overrightarrow{J_f} - \rho_f \overrightarrow{v} \big[\big] + \nabla_{\Sigma} \cdot \overrightarrow{K_f} = -\frac{\partial{\sigma_f}}{\partial{t}} \nonumber \] | \[ \overrightarrow{n} \times \big[\big] \overrightarrow{E} + v \times \mu_o \overrightarrow{H} \big[\big] = 0 \label{23} \] |
| \[ \overrightarrow{n} \times \big[\big] \overrightarrow{H} - v \times \varepsilon_o \overrightarrow{E} \big[\big] = K_f - \sigma_f \overrightarrow{v_t} \nonumber \] | \[ \overrightarrow{n} \cdot \big[\big] \overrightarrow{J_f} \big[\big] = 0 \label{24} \] |
Included in the summary are several that are either rarely used, are matters of definition or are obvious. That the surface polarization charge and surface magnetic charge are related to f and A respectively follows from Eqs. 2.8.3 and 2.9.2 used in conjunction with Gauss' theorem and the elemental volume of Figure 2.10.1. Similarly, Equation \ref{24}b follows from the solenoidal nature of the MQS current density. Finally, Equation \ref{24}a follows from the EQS form of Ampere's law, integrated over the surface \(S\) of Figure 2.10.3, following the line of reasoning used in connection with Equation \ref{20}.