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 ($latex \vec{E} ,\vec{D}$) and magnetic fields ($latex \vec{B} ,\vec{H}$) to their respective sources – charge density ($latex \rho $) and current density ( $latex \vec{J} $ ).
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.
$latex \begin{matrix} \text{Differential form} &&& \text{Integral form} \\ \bigtriangledown \cdot \vec{D} = \rho & & & \displaystyle{\oint_{S}\vec{D} \cdot d\vec{S} = \int_{V} \rho \; dV} \\ \bigtriangledown \cdot \vec{B} = 0 & & & \displaystyle{\oint_{S}\vec{B} \cdot d\vec{S} = 0} \\ \bigtriangledown \times \vec{E} = – \frac{\partial \vec{B}}{\partial t} & & & \displaystyle{\oint_{C}\vec{E} \cdot d\vec{l} = – \frac{\partial}{\partial t} \int_{S} \vec{B} \cdot d \vec{S} } \\ \bigtriangledown \times \vec{H} = \vec{J} + \frac{\vec{D}}{\partial t} & & & \displaystyle{\oint_{C}\vec{H} \cdot d\vec{l} = \int_{S} \vec{J} \cdot d\vec{S} + \frac{\partial}{\partial t}\int_{S} \vec{D} \cdot d \vec{S}} \end{matrix} $
Maxwell’s equation (1):
$latex \boxed{ \displaystyle{\oint_{S}\vec{D} \cdot d\vec{S} = \int_{V} \rho \; dV} } &s=2 $
The flux of the displacement electric field $latex \vec{D}$ through a closed surface $latex S$ equals the total electric charge enclosed in the corresponding volume space $latex V$.
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 $latex 4 \pi r^2$.
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 $latex \vec{D}$.
$latex \displaystyle{ \oint_{S} \vec{D} \cdot d\vec{S} = 4 \pi r^2 \; |\vec{D}| = 4 \pi r^2 \; D} \;\;\;\; (5)$
For the right hand side of the Maxwell’s equation (1), the integral of the charge density $latex \rho$ over a volume V is simply equal to the charge enclosed. Therefore,
$latex \displaystyle{ \int_{V} \rho \; dV = q} \;\;\;\; (6) $
The electric displacement field $latex \vec{D}$ is a measure of electric field in the material after taking into account the effect of the material on the electric field. The electric field $latex \vec{E}$ and the displacement field $latex \vec{D}$ are related by the permittivity of the material $latex \epsilon$ as
$latex \vec{D} = \epsilon \vec{E} \;\;\;\; (7) $
Combining equations (5), (6) and (7), yields the magnitude of an electric field as derived from Coulomb’s law
$latex \displaystyle{ E = \frac{q}{ 4 \pi \epsilon r^2}} \;\;\;\; (8)$
Maxwell’s equation (2)
$latex \boxed{ \displaystyle{\oint_{S}\vec{B} \cdot d\vec{S} = 0} } &s=2 $
The flux of the magnetic field $latex \vec{B}$ 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)
$latex \boxed{\displaystyle{\oint_{C}\vec{E} \cdot d\vec{l} = – \frac{\partial}{\partial t} \int_{S} \vec{B} \cdot d \vec{S} }} &s=2 $
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.
$latex emf = \displaystyle{\oint_{C}\vec{E} \cdot d\vec{l} } \;\;\;\; (9) $
The right side of the equation is the rate of change of magnetic flux through the circuit.
$latex \displaystyle{\frac{\partial}{\partial t} \int_{S} \vec{B} \cdot d \vec{S} = \frac{\partial \Phi }{\partial t}} \;\;\;\; (10) $
Hence, the Maxwell’s third equation is actually the Faraday’s (and Len’s) law of magnetic induction
$latex \boxed{\displaystyle{\oint_{C}\vec{E} \cdot d\vec{l} = emf = – \frac{\partial}{\partial t} \int_{S} \vec{B} \cdot d \vec{S} =- \frac{\partial \Phi }{\partial t} } } &s=2 \;\;\;\; (11) &s=2 $
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)
$latex \boxed{\displaystyle{\int_{S} \vec{J} \cdot d\vec{S} + \frac{\partial}{\partial t}\int_{S} \vec{D} \cdot d \vec{S} = \oint_{C}\vec{H} \cdot d\vec{l}}} &s=2$
The circulating magnetic field is denoted by the circulation of magnetizing field $latex \vec{H}$ around a closed curved path : $latex \oint_{C}\vec{H} \cdot d\vec{l} $. The electric current is denoted by the flux of current density ($latex J$) through any surface spanning that curved path. The quantity $latex \frac{\partial}{\partial t}\int_{S} \vec{D} \cdot d \vec{S} $ denotes the rate of change of displacement current $latex \vec{D}$ through any surface spanning that curved path.
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 ↗
Books by the author
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