Key focus: Briefly look at the building blocks of antenna array theory starting from the fundamental Maxwell’s equations in electromagnetism.
Maxwell’s equations
Maxwell’s equations are a collection of equations that describe the behavior of electromagnetic fields. The equations relate the electric fields (
Maxwell’s equations are available in two forms: differential form and integral form. The integral forms of Maxwell’s equations are helpful in their understanding the physical significance.
Maxwell’s equation (1):
The flux of the displacement electric field
through a closed surface equals the total electric charge enclosed in the corresponding volume space .
This is also called Gauss law for electricity.
Consider a point charge +q in a three dimensional space. Assuming a symmetric field around the charge and at a distance r from the charge, the surface area of the sphere is
Therefore, left side of the equation is simply equal to the surface area of the sphere multiplied by the magnitude of the electric displacement vector
For the right hand side of the Maxwell’s equation (1), the integral of the charge density
The electric displacement field
Combining equations (5), (6) and (7), yields the magnitude of an electric field as derived from Coulomb’s law
Maxwell’s equation (2)
The flux of the magnetic field
through a closed surface is zero. That is, the net of magnetic field that “flows into” and “flows out of” a closed surface is zero.
This implies that there are no source or sink for the magnetic flux lines, in other words – they are closed field lines with no beginning or end. This is also called Gauss law for magnetic field.
Maxwell’s equation (3)
The work done on an electric charge as it travels around a closed loop conductor is the electromotive force (emf). Therefore, the left side of the gives the emf induced in a circuit.
The right side of the equation is the rate of change of magnetic flux through the circuit.
Hence, the Maxwell’s third equation is actually the Faraday’s (and Len’s) law of magnetic induction
The electromotive force (emf) induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.
Maxwell’s equation (4)
The circulating magnetic field is denoted by the circulation of magnetizing field
According to Maxwell’s extension to the Ampere’s law , magnetic fields can be generated in two ways: with electric current and with changing electric flux. The equation states that the electric current or change in electric flux through a surface produces a circulating magnetic field around any path that bounds that surface.
Summary of Maxwell’s equations
The electric field leaving a volume space is proportional to the electric charge contained in the volume.
The net of magnetic field that “flows into” and “flows out of” a closed surface is zero. There is no concept called magnetic charge/magnetic monopole.
A changing magnetic flux through a circuit induces electromotive force in the circuit
Magnetic fields are produced by electric current as well as by changing electric flux.
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References
[1] The Feynman lectures on physics – online edition ↗
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