From Maxwell’s equations to antenna array – part 1

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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 (\vec{E} ,\vec{D}) and magnetic fields (\vec{B} ,\vec{H}) to their respective sources – charge density (\rho ) and current density ( \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.

\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):

\boxed{ \displaystyle{\oint_{S}\vec{D} \cdot d\vec{S} = \int_{V} \rho \; dV} }

The flux of the displacement electric field \vec{D} through a closed surface S equals the total electric charge enclosed in the corresponding volume space 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 4 \pi r^2.

Illustration of Coulomb's law using Maxwell's equation
Figure 1: Illustration of Coulomb’s law using Maxwell’s equation

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 \vec{D}.

\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 \rho over a volume V is simply equal to the charge enclosed. Therefore,

\displaystyle{ \int_{V} \rho \; dV = q} \;\;\;\; (6)

The electric displacement field \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 \vec{E} and the displacement field \vec{D} are related by the permittivity of the material \epsilon as

\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

\displaystyle{ E = \frac{q}{ 4 \pi \epsilon r^2}} \;\;\;\; (8)

Maxwell’s equation (2)

\boxed{ \displaystyle{\oint_{S}\vec{B} \cdot d\vec{S} = 0} }

The flux of the magnetic field \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.

Gauss law for magnetic field
Figure 2: Gauss law for magnetic field

Maxwell’s equation (3)

\boxed{\displaystyle{\oint_{C}\vec{E} \cdot d\vec{l} = - \frac{\partial}{\partial t} \int_{S} \vec{B} \cdot d \vec{S} }}

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.

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.

\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

\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} } } \;\;\;\; (11)

The electromotive force (emf) induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.

Faradays law for magnetic induction emf change of magnetic flux
Figure 3: Faraday’s law for magnetic induction

Maxwell’s equation (4)

\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}}}

The circulating magnetic field is denoted by the circulation of magnetizing field \vec{H} around a closed curved path : \oint_{C}\vec{H} \cdot d\vec{l} . The electric current is denoted by the flux of current density (J) through any surface spanning that curved path. The quantity \frac{\partial}{\partial t}\int_{S} \vec{D} \cdot d \vec{S} denotes the rate of change of displacement current \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 ↗

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