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2.15: Solenoidal Fields, Vector Potential and Stream Function

  • Page ID
    31787
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    Irrotational fields, such as the quasistatic electric field, are naturally represented by a scalar potential. Not only does this reduce the vector field to a scalar field, but the potential function evaluated on such surfaces as those of "perfectly" conducting electrodes becomes a lumped parameter terminal variable, e.g., the voltage.

    Solenoidal fields, such as the magnetic flux density \(\overrightarrow{B}\), are for similar reasons sometimes represented in terms of a vector potential \(\overrightarrow{A}\):

    \[\overrightarrow{B} = \nabla \times \overrightarrow{A} \label{1} \]

    Thus, \(\overrightarrow{B}\) automatically has no divergence. Unfortunately, the vector field \(\overrightarrow{B}\) is represented in terms of another vectox field \(\overrightarrow{A}\). However, for important two-dimensional or symmetric configurations, a single component of \(\overrightarrow{A}\) is all required to again reduce the description to one involving a scalar function. Four commonly encountered cases are summarized in-Table 2.18.1.

    The first two are two-dimensional in the usual sense. The field \(\overrightarrow{B}\) lies in the \(x-y\) \((or\, r-\theta)\) plane and depends only on these coordinates. The associated vector potential has only a \(z\) component. The third configuration, like the second, is in cylindrical geometry, but with \(\overrightarrow{B}\) independent of \(\theta\) and hence with \(\overrightarrow{A}\) having only an \(\overrightarrow{i}_{\theta}\) component. The fourth configuration is in spherical geometry with symmetry about the \(z\) axis and the vector potential directed along \(\phi_{\theta}\).

    Like the scalar potential used to represent irrotational fields, the vector potential is closely related to lumped parameter variables. If \(\overrightarrow{B}\) is the magnetic flux density, it is convenient for evaluation of the flux linkage \(\lambda\) (Equation 2.12.1). For an incompressible flow, where \(\overrightarrow{B}\) is replaced by the fluid velocity \(\overrightarrow{v}\), the vector potential is conveniently used to evaluate the volume rate of flow. In that application, \(A\) and \(\Lambda\) become "stream functions."

    The connection between the flux linked and the vector potential follows from Stokes's theorem, Equation 2.6.3. The flux \(\phi_{\lambda}\) through a surface \(S\) enclosed by a contour \(C\) is

    \[ \phi_{\lambda} = \int_S \overrightarrow{B} \cdot \overrightarrow{n} da = \int_S \nabla \times \overrightarrow{A} \cdot \overrightarrow{n} da = \oint_C \overrightarrow{A} \cdot d \overrightarrow{l} \label{2} \]

    In each of the configurations of Table 2.18.1, Equation \ref{2} amounts to an evaluation of the surface integral. For example, in the Cartesian two-dimensional configuration, contributions to the integration around a contour \(C\) enclosing a surface having length \(l\) in the \(z\) direction, only come from the legs running in the a direction. Along these portions of the contour, denoted by (a) and (b), the coordinates \((x,y)\) are constant. Hence, the flux through the surface is simply \(l\) times the difference \(A(a) - A(b)\), as summarized in Table 2.18.1.

    In the axisymmetric cylindrical and spherical configurations, \(r\) and \(r\, sin \theta)\) dependences are respectively introduced, so that evaluation of \(\Lambda\) essentially gives the flux linked. For example, in the spherical configuration, the flux linked by a surface having inner and outer radii \(r\, cos \theta)\) evaluated at (a) and (b) is simply

    \[ \phi_{\lambda} = \oint_C \frac{\Lambda (r, \theta)}{r\, sin \theta} \overrightarrow{i}_{\phi} \cdot d \overrightarrow{l} = \frac{\Lambda}{r\, sin \theta} 2 \pi ( r \, sin \theta) \bigg \vert_b^a = 2 \pi [ \Lambda(a) - \Lambda(b)] \label{3} \]

    Used in fluid mechanics to represent incompressible fluid flow, \(\Lambda\) is the Stokel's stream function. Note that the flux is positive if directed through the surface in the direction of \(\overrightarrow{n}\), which is specified in terms of the contour \(C\) by the right-hand rule.


    This page titled 2.15: Solenoidal Fields, Vector Potential and Stream Function is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.