Lecture 1

Introduction, Maxwell’s

Equations

1.1 Importance of Electromagnetics

We will explain why electromagnetics is so important, and its impact on very many different

areas. Then we will give a brief history of electromagnetics, and how it has evolved in the

modern world. Then we will go briefly over Maxwell’s equations in their full glory. But we

will begin the study of electromagnetics by focussing on static problems.

The discipline of electromagnetic field theory and its pertinent technologies is also known

as electromagnetics. It has been based on Maxwell’s equations, which are the result of the

seminal work of James Clerk Maxwell completed in 1865, after his presentation to the British

Royal Society in 1864. It has been over 150 years ago now, and this is a long time compared

to the leaps and bounds progress we have made in technological advancements. But despite,

research in electromagnetics has continued unabated despite its age. The reason is that

electromagnetics is extremely useful, and has impacted a large sector of modern technologies.

To understand why electromagnetics is so useful, we have to understand a few points

about Maxwell’s equations.

First, Maxwell’s equations are valid over a vast length scale from subatomic dimensions

to galactic dimensions. Hence, these equations are valid over a vast range of wavelengths,

going from static to ultra-violet wavelengths.1

Maxwell’s equations are relativistic invariant in the parlance of special relativity [1]. In

fact, Einstein was motivated with the theory of special relativity in 1905 by Maxwell’s

equations [2]. These equations look the same, irrespective of what inertial reference

frame one is in.

Maxwell’s equations are valid in the quantum regime, as it was demonstrated by Paul

Dirac in 1927 [3]. Hence, many methods of calculating the response of a medium to

1Current lithography process is working with using ultra-violet light with a wavelength of 193 nm.

1

2 Electromagnetic Field Theory

classical field can be applied in the quantum regime also. When electromagnetic theory

is combined with quantum theory, the field of quantum optics came about. Roy Glauber

won a Nobel prize in 2005 because of his work in this area [4].

Maxwell’s equations and the pertinent gauge theory has inspired Yang-Mills theory

(1954) [5], which is also known as a generalized electromagnetic theory. Yang-Mills

theory is motivated by differential forms in differential geometry [6]. To quote from

Misner, Thorne, and Wheeler, “Differential forms illuminate electromagnetic theory,

and electromagnetic theory illuminates differential forms.” [7, 8]

Maxwell’s equations are some of the most accurate physical equations that have been

validated by experiments. In 1985, Richard Feynman wrote that electromagnetic theory

has been validated to one part in a billion.2 Now, it has been validated to one part in

a trillion (Aoyama et al, Styer, 2012).3

As a consequence, electromagnetics has had a tremendous impact in science and tech-

nology. This is manifested in electrical engineering, optics, wireless and optical commu-

nications, computers, remote sensing, bio-medical engineering etc.

Figure 1.1: The impact of electromagnetics in many technologies. The areas in blue are

prevalent areas impacted by electromagnetics some 20 years ago [9], and the areas in red are

modern emerging areas impacted by electromagnetics.

2This means that if a jet is to fly from New York to Los Angeles, an error of one part in a billion means

an error of a few millmeters.

3This means an error of a hairline, if one were to fly from the earth to the moon.

Introduction, Maxwell’s Equations 3

1.2 A Brief History of Electromagnetics

Electricity and magnetism have been known to humans for a long time. Also, the physical

properties of light has been known. But electricity and magnetism, now termed electromag-

netics in the modern world, has been thought to be governed by different physical laws as

opposed to optics. This is understandable as the physics of electricity and magnetism is quite

different of the physics of optics as they were known to humans.

For example, lode stone was known to the ancient Greek and Chinese around 600 BC

to 400 BC. Compass was used in China since 200 BC. Static electricity was reported by

the Greek as early as 400 BC. But these curiosities did not make an impact until the age

of telegraphy. The coming about of telegraphy was due to the invention of the voltaic cell

or the galvanic cell in the late 1700’s, by Luigi Galvani and Alesandro Volta [10]. It was

soon discovered that two pieces of wire, connected to a voltaic cell, can be used to transmit

information.

So by the early 1800’s this possibility had spurred the development of telegraphy. Both

Andr-Marie Ampre (1823) [11, 12] and Michael Faraday (1838) [13] did experiments to bet-

ter understand the properties of electricity and magnetism. And hence, Ampere’s law and

Faraday law are named after them. Kirchhoff voltage and current laws were also developed

in 1845 to help better understand telegraphy [14, 15]. Despite these laws, the technology of

telegraphy was poorly understood. It was not known as to why the telegraphy signal was

distorted. Ideally, the signal should be a digital signal switching between one’s and zero’s,

but the digital signal lost its shape rapidly along a telegraphy line.4

It was not until 1865 that James Clerk Maxwell [17] put in the missing term in Ampere’s

law, the term that involves displacement current, only then the mathematical theory for

electricity and magnetism was complete. Ampere’s law is now known as generalized Ampere’s

law. The complete set of equations are now named Maxwell’s equations in honor of James

Clerk Maxwell.

The rousing success of Maxwell’s theory was that it predicted wave phenomena, as they

have been observed along telegraphy lines. Heinrich Hertz in 1888 [18] did experiment to

proof that electromagnetic field can propagate through space across a room. Moreover, from

experimental measurement of the permittivity and permeability of matter, it was decided

that electromagnetic wave moves at a tremendous speed. But the velocity of light has been

known for a long while from astronomical observations (Roemer, 1676) [19]. The observation

of interference phenomena in light has been known as well. When these pieces of information

were pieced together, it was decided that electricity and magnetism, and optics, are actually

governed by the same physical law or Maxwell’s equations. And optics and electromagnetics

are unified into one field.

4As a side note, in 1837, Morse invented the Morse code for telegraphy [16]. There were cross pollination

of ideas across the Atlantic ocean despite the distance. In fact, Benjamin Franklin associated lightning with

electricity in the latter part of the 18-th century. Also, notice that electrical machinery was invented in 1832

even though electromagnetic theory was not fully understood.

4 Electromagnetic Field Theory

Figure 1.2: A brief history of electromagnetics and optics as depicted in this figure.

In Figure 1.2, a brief history of electromagnetics and optics is depicted. In the beginning,

it was thought that electricity and magnetism, and optics were governed by different physical

laws. Low frequency electromagnetics was governed by the understanding of fields and their

interaction with media. Optical phenomena were governed by ray optics, reflection and

refraction of light. But the advent of Maxwell’s equations in 1865 reveal that they can be

unified by electromagnetic theory. Then solving Maxwell’s equations becomes a mathematical

endeavor.

The photo-electric effect [20, 21], and Planck radiation law [22] point to the fact that

electromagnetic energy is manifested in terms of packets of energy. Each unit of this energy

is now known as the photon. A photon carries an energy packet equal to ℏω, where ω is the

angular frequency of the photon and ℏ = 6.626 × 10−34 J s, the Planck constant, which is

a very small constant. Hence, the higher the frequency, the easier it is to detect this packet

of energy, or feel the graininess of electromagnetic energy. Eventually, in 1927 [3], quantum

theory was incorporated into electromagnetics, and the quantum nature of light gives rise to

the field of quantum optics. Recently, even microwave photons have been measured [23]. It

is a difficult measurement because of the low frequency of microwave (109 Hz) compared to

optics (1015 Hz): microwave photon has a packet of energy about a million times smaller than

that of optical photon.

The progress in nano-fabrication [24] allows one to make optical components that are

subwavelength as the wavelength of blue light is about 450 nm. As a result, interaction of

light with nano-scale optical components requires the solution of Maxwell’s equations in its

full glory.

Introduction, Maxwell’s Equations 5

In 1980s, Bell’s theorem (by John Steward Bell) [25] was experimentally verified in favor of

the Copenhagen school of quantum interpretation (led by Niel Bohr) [26]. This interpretation

says that a quantum state is in a linear superposition of states before a measurement. But

after a measurement, a quantum state collapses to the state that is measured. This implies

that quantum information can be hidden in a quantum state. Hence, a quantum particle,

such as a photon, its state can remain incognito until after its measurement. In other words,

quantum theory is “spooky”. This leads to growing interest in quantum information and

quantum communication using photons. Quantum technology with the use of photons, an

electromagnetic quantum particle, is a subject of growing interest.

1.3 Maxwell’s Equations in Integral Form

Maxwell’s equations can be presented as fundamental postulates.5 We will present them in

their integral forms, but will not belabor them until later.

˛

C

E · dl = − d

dt

¨

S

B · dS Faraday’s Law (1.3.1)

˛

C

H · dl = d

dt

¨

S

D · dS + I Ampere’s Law (1.3.2)

‹

S

D · dS = Q Gauss’s or Coulomb’s Law (1.3.3)

‹

S

B · dS = 0 Gauss’s Law (1.3.4)

The units of the basic quantities above are given as:

E: V/m H: A/m

D: C/m2 B: Webers/m2

I: A Q: Coulombs

5Postulates in physics are similar to axioms in mathematics. They are assumptions that need not be

proved.

6 Electromagnetic Field Theory

1.4 Coulomb’s Law (Statics)

This law, developed in 1785 [27], expresses the force between two charges q1 and q2. If these

charges are positive, the force is repulsive and it is given by

f1→2 = q1q2

4πεr2 ˆr12 (1.4.1)

Figure 1.3: The force between two charges q1 and q2. The force is repulsive if the two charges

have the same sign.

f (force): newton

q (charge): coulombs

ε (permittivity): farads/meter

r (distance between q1 and q2): m

ˆr12= unit vector pointing from charge 1 to charge 2

ˆr12 = r2 − r1

|r2 − r1|, r = |r2 − r1| (1.4.2)

Since the unit vector can be defined in the above, the force between two charges can also be

rewritten as

f1→2 = q1q2(r2 − r1)

4πε|r2 − r1|3 , (r1, r2 are position vectors) (1.4.3)

Introduction, Maxwell’s Equations 7

1.5 Electric Field E (Statics)

The electric field E is defined as the force per unit charge [28]. For two charges, one of charge

q and the other one of incremental charge ∆q, the force between the two charges, according

to Coulomb’s law (1.4.1), is

f = q∆q

4πεr2 ˆr (1.5.1)

where ˆr is a unit vector pointing from charge q to the incremental charge ∆q. Then the force

per unit charge is given by

E = f

4q , (V/m) (1.5.2)

This electric field E from a point charge q at the orgin is hence

E = q

4πεr2 ˆr (1.5.3)

Therefore, in general, the electric field E(r) from a point charge q at r′ is given by

E(r) = q(r − r′)

4πε|r − r′|3 (1.5.4)

where

ˆr = r − r′

|r − r′| (1.5.5)

Figure 1.4: Emanating E field from an electric point charge as depicted by depicted by (1.5.4)

and (1.5.3).

8 Electromagnetic Field Theory

Example 1

Field of a ring of charge of density %l C/m

Figure 1.5: Electric field of a ring of charge (Courtesy of Ramo, Whinnery, and Van Duzer)

[29].

Question: What is E along z axis?

Remark: If you know E due to a point charge, you know E due to any charge distribution

because any charge distribution can be decomposed into sum of point charges. For instance, if

there are N point charges each with amplitude qi, then by the principle of linear superposition,

the total field produced by these N charges is

E(r) =

N ∑

i=1

qi(r − ri)

4πε|r − ri|3 (1.5.6)

where qi = %(ri)∆Vi. In the continuum limit, one gets

E(r) =

ˆ

V

%(r′)(r − r′)

4πε|r − r′|3 dV (1.5.7)

In other words, the total field, by the principle of linear superposition, is the integral sum-

mation of the contributions from the distributed charge density %(r).

Introduction, Maxwell’s Equations 9

1.6 Gauss’s Law (Statics)

This law is also known as Coulomb’s law as they are closely related to each other. Apparently,

this simple law was first expressed by Joseph Louis Lagrange [30] and later, reexpressed by

Gauss in 1813 (wikipedia).

This law can be expressed as ‹

S

D · dS = Q (1.6.1)

D: electric flux density C/m2 D = εE.

dS: an incremental surface at the point on S given by dS ˆn where ˆn is the unit normal

pointing outward away from the surface.

Q: total charge enclosed by the surface S.

Figure 1.6: Electric flux (Courtesy of Ramo, Whinnery, and Van Duzer) [29]

The left-hand side of (1.6.1) represents a surface integral over a closed surface S. To

understand it, one can break the surface into a sum of incremental surfaces ∆Si, with a

local unit normal ˆni associated with it. The surface integral can then be approximated by a

summation ‹

S

D · dS ≈ ∑

i

Di · ˆni∆Si = ∑

i

Di · ∆Si (1.6.2)

where one has defined ∆Si = ˆni∆Si. In the limit when ∆Si becomes infinitesimally small,

the summation becomes a surface integral.

1.7 Derivation of Gauss’s Law from Coulomb’s Law (Stat-

ics)

From Coulomb’s law and the ensuing electric field due to a point charge, the electric flux is

D = εE = q

4πr2 ˆr (1.7.1)

10 Electromagnetic Field Theory

When a closed spherical surface S is drawn around the point charge q, by symmetry, the

electric flux though every point of the surface is the same. Moreover, the normal vector ˆn

on the surface is just ˆr. Consequently, D · ˆn = D · ˆr = q/(4πr2), which is a constant on a

spherical of radius r. Hence, we conclude that for a point charge q, and the pertinent electric

flux D that it produces on a spherical surface,

‹

S

D · dS = 4πr2D · ˆn = q (1.7.2)

Therefore, Gauss’s law is satisfied by a point charge.

Figure 1.7: Electric flux from a point charge satisfies Gauss’s law.

Even when the shape of the spherical surface S changes from a sphere to an arbitrary

shape surface S, it can be shown that the total flux through S is still q. In other words, the

total flux through sufaces S1 and S2 in Figure 1.8 are the same.

This can be appreciated by taking a sliver of the angular sector as shown in Figure 1.9.

Here, ∆S1 and ∆S2 are two incremental surfaces intercepted by this sliver of angular sector.

The amount of flux passing through this incremental surface is given by dS · D = ˆn · D∆S =

ˆn · ˆrDr ∆S. Here, D = ˆrDr is pointing in the ˆr direction. In ∆S1, ˆn is pointing in the ˆr

direction. But in ∆S2, the incremental area has been enlarged by that ˆn not aligned with

D. But this enlargement is compensated by ˆn · ˆr. Also, ∆S2 has grown bigger, but the flux

at ∆S2 has grown weaker by the ratio of (r2/r1)2. Finally, the two fluxes are equal in the

limit that the sliver of angular sector becomes infinitesimally small. This proves the assertion

that the total fluxes through S1 and S2 are equal. Since the total flux from a point charge q

through a closed surface is independent of its shape, but always equal to q, then if we have a

total charge Q which can be expressed as the sum of point charges, namely.

Q = ∑

i

qi (1.7.3)

Then the total flux through a closed surface equals the total charge enclosed by it, which is

the statement of Gauss’s law or Coulomb’s law.

Example 2

Introduction, Maxwell’s Equations 11

Figure 1.8: Same amount of electric flux from a point charge passes through two surfaces S1

and S2.

Figure 1.9: When a sliver of angular sector is taken, same amount of electric flux from a point

charge passes through two incremental surfaces ∆S1 and ∆S2.

12 Electromagnetic Field Theory

Figure 1.10: Figure for Example 2 for a coaxial cylinder.

Field between coaxial cylinders of unit length.

Question: What is E?

Hint: Use symmetry and cylindrical coordinates to express E = ˆρEρ and appply Gauss’s law.

Introduction, Maxwell’s Equations 13

Example 3:

Fields of a sphere of uniform charge density.

Figure 1.11: Figure for Example 3 for a sphere with uniform charge density.

Question: What is E?

Hint: Again, use symmetry and spherical coordinates to express E = ˆrEr and appply Gauss’s

law.

14 Electromagnetic Field Theory

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