Electric-field integral equation

The electric-field integral equation is a relationship that allows the calculation of an electric field (E) generated by an electric current distribution (J).

Derivation

When all quantities in the frequency domain are considered, a time-dependency $e^{jwt}$ that is suppressed throughout is assumed.

Beginning with the Maxwell equations relating the electric and magnetic field, and assuming a linear, homogeneous media with permeability $\mu$ and permittivity $\varepsilon \,$: $${\begin{aligned}\nabla \times \mathbf {E} &=-j\omega \mu \mathbf {H} \\[1ex]\nabla \times \mathbf {H} &=j\omega \varepsilon \mathbf {E} +\mathbf {J} \end{aligned}}$$

Following the third equation involving the divergence of H $$\nabla \cdot \mathbf {H} =0\,$$ by vector calculus we can write any divergenceless vector as the curl of another vector, hence $$\nabla \times \mathbf {A} =\mathbf {H}$$ where A is called the magnetic vector potential. Substituting this into the above we get $$\nabla \times (\mathbf {E} +j\omega \mu \mathbf {A} )=0$$ and any curl-free vector can be written as the gradient of a scalar, hence $$\mathbf {E} +j\omega \mu \mathbf {A} =-\nabla \Phi$$ where $\Phi$ is the electric scalar potential. These relationships now allow us to write $$\nabla \times \nabla \times \mathbf {A} -k^{2}\mathbf {A} =\mathbf {J} -j\omega \varepsilon \nabla \Phi$$ where $k=\omega {\sqrt {\mu \varepsilon }}$, which can be rewritten by vector identity as $$\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} -k^{2}\mathbf {A} =\mathbf {J} -j\omega \varepsilon \nabla \Phi$$

As we have only specified the curl of A, we are free to define the divergence, and choose the following: $$\nabla \cdot \mathbf {A} =-j\omega \varepsilon \Phi \,$$ which is called the Lorenz gauge condition. The previous expression for A now reduces to $$\nabla ^{2}\mathbf {A} +k^{2}\mathbf {A} =-\mathbf {J} \,$$ which is the vector Helmholtz equation. The solution of this equation for A is $$\mathbf {A} (\mathbf {r} )={\frac {1}{4\pi }}\int \mathbf {J} (\mathbf {r} ^{\prime })\ G(\mathbf {r} ,\mathbf {r} ^{\prime })\,d\mathbf {r} ^{\prime }$$ where $G(\mathbf {r} ,\mathbf {r} ^{\prime })$ is the three-dimensional homogeneous Green's function given by $$G(\mathbf {r} ,\mathbf {r} ^{\prime })={\frac {e^{-jk\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}$$

We can now write what is called the electric field integral equation (EFIE), relating the electric field E to the vector potential A $$\mathbf {E} =-j\omega \mu \mathbf {A} +{\frac {1}{j\omega \varepsilon }}\nabla (\nabla \cdot \mathbf {A} )\,$$

We can further represent the EFIE in the dyadic form as $$\mathbf {E} =-j\omega \mu \int _{V}d\mathbf {r} ^{\prime }\mathbf {G} (\mathbf {r} ,\mathbf {r} ^{\prime })\cdot \mathbf {J} (\mathbf {r} ^{\prime })\,$$ where $\mathbf {G} (\mathbf {r} ,\mathbf {r} ^{\prime })\,$ here is the dyadic homogeneous Green's Function given by $$\mathbf {G} (\mathbf {r} ,\mathbf {r} ^{\prime })={\frac {1}{4\pi }}\left[\mathbf {I} +{\frac {\nabla \nabla }{k^{2}}}\right]G(\mathbf {r} ,\mathbf {r} ^{\prime })$$

Interpretation

The EFIE describes a radiated field E given a set of sources J, and as such it is the fundamental equation used in antenna analysis and design. It is a very general relationship that can be used to compute the radiated field of any sort of antenna once the current distribution on it is known. The most important aspect of the EFIE is that it allows us to solve the radiation/scattering problem in an unbounded region, or one whose boundary is located at infinity. For closed surfaces, it is possible to use the Magnetic Field Integral Equation or the Combined Field Integral Equation, both of which result in a set of equations with improved condition number compared to the EFIE. However, the MFIE and CFIE can still contain resonances.

In scattering problems, it is desirable to determine an unknown scattered field $E_{s}$ that is due to a known incident field $E_{i}$. Unfortunately, the EFIE relates the scattered field to J, not the incident field, so we do not know what J is. This sort of problem can be solved by imposing the boundary conditions on the incident and scattered field, allowing one to write the EFIE in terms of $E_{i}$ and J alone. Once this has been done, the integral equation can then be solved by a numerical technique appropriate to integral equations such as the method of moments.

Notes

By the Helmholtz theorem a vector field is described completely by its divergence and curl. As the divergence was not defined, we are justified by choosing the Lorenz Gauge condition above provided that we consistently use this definition of the divergence of A in all subsequent analysis. However, other choices for $\nabla \cdot \mathbf {A}$ are just as valid and lead to other equations, which all describe the same phenomena, and the solutions of the equations for any choice of $\nabla \cdot \mathbf {A}$ lead to the same electromagnetic fields, and the same physical predictions about the fields and charges are accelerated by them.

It is natural to think that if a quantity exhibits this degree of freedom in its choice, then it should not be interpreted as a real physical quantity. After all, if we can freely choose $\nabla \cdot \mathbf {A}$ to be anything, then $\mathbf {A}$ is not unique. One may ask: what is the "true" value of $\mathbf {A}$ measured in an experiment? If $\mathbf {A}$ is not unique, then the only logical answer must be that we can never measure the value of $\mathbf {A}$. On this basis, it is often stated that it is not a real physical quantity and it is believed that the fields $\mathbf {E}$ and $\mathbf {B}$ are the true physical quantities.

However, there is at least one experiment in which value of the $\mathbf {E}$ and $\mathbf {B}$ are both zero at the location of a charged particle, but it is nevertheless affected by the presence of a local magnetic vector potential; see the Aharonov–Bohm effect for details. Nevertheless, even in the Aharonov–Bohm experiment, the divergence $\mathbf {A}$ never enters the calculations; only $\nabla \times \mathbf {A}$ along the path of the particle determines the measurable effect.

References

• Gibson, Walton C. The Method of Moments in Electromagnetics. Chapman & Hall/CRC, 2008. ISBN 978-1-4200-6145-1
• Harrington, Roger F. Time-Harmonic Electromagnetic Fields. McGraw-Hill, Inc., 1961. ISBN 0-07-026745-6.
• Balanis, Constantine A. Advanced Engineering Electromagnetics. Wiley, 1989. ISBN 0-471-62194-3.
• Chew, Weng C. Waves and Fields in Inhomogeneous Media. IEEE Press, 1995. ISBN 0-7803-4749-8.
• Rao, Wilton, Glisson. Electromagnetic Scattering by Surfaces of Arbitrary Shape. IEEE Transactions on Antennas and Propagation, vol, AP-30, No. 3, May 1982. doi:10.1109/TAP.1982.1142818