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Home , James Clerk Maxwell

Maxwell Equations and Hyperbolic

by KWAN NGA WONG 17220955

A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of (Honours) in 2021

at Hong Kong Baptist University

29 December 2020 Acknowledgement

The thesis was my original and carried out under the supervision of Dr. Wu, Wei. I would like to express my sincere gratitude to Dr. Wu for his guidance and encouragement throughout the whole process. In every meeting, I learned new knowledge regarding the topic, and Dr. Wu gave distinct instructions and remarks for my work. I would like to thank Dr. Wu for providing supports enthusiastically which ease my enquiries and nerves. I would also like to thank Dr. Zhuang, Lina for being my observer.

WONG KWAN NGA

Department of

Hong Kong Baptist University

December 29, 2020

ii Abstract

In this project, we are going to introduce ’s equation which merged the concepts of ’s law, Faraday’s law, and ’s law. Maxwell’s equation discovered the relationship between electric field and magnetic field, we are going to provide a full derivation for the four Maxwell’s equations including both integral form and differential form in mathematical aspects. Moreover, we are going to investigate the simplified form of Maxwell’s equations in some special cases and how it can be expressed under boundary conditions. We will also briefly introduce and look into the mathematical formulation of metamaterial, and its special form, hyperbolic metamaterial.

iii Contents

1 Maxwell Equations2 1.1 Introduction ...... 2 1.2 Gauss’ Law for ...... 3 1.3 Gauss’ Law for ...... 8 1.4 Faraday’s Law of Induction ...... 10 1.5 Ampere’s Law ...... 12

2 Special Cases and Boundary Conditions 16 2.1 Electromagnetic in Vacuum ...... 16 2.2 Time Harmonic ...... 17 2.3 Boundary Conditions ...... 18

3 Hyperbolic metamaterial 22

4 Conclusion 25

Bibliography 26

1 1 Maxwell Equations

1.1 Introduction

Electromagnetic are used in cellphone, x-ray, body, and baggage scanners, which brings convenience to our daily life. To start with, we could explore the history of re- search on and electromagnetic waves[14, 6, 8].

In the 17th century, Issac explore that light are a stream of discrete tiny par- ticles, any luminous source amidst the corpuscles which travel in a straight line in all directions. But in Newton’s corpuscular theory of light, he fails to explain the interfer- ence and of light. In a later time, Huygen’s wave theory proves that light is a wave that successfully explains the phenomena of light.

In 1865, James Clerk Maxwell has introduced the dynamical theory of electromagnetic waves; Maxwell infers that light is an electromagnetic wave by deriving the electromag- netic wave equation with the relationship of the . In details, Maxwell deduced that electric field and magnetic field are waves which oscillate in a different direction, and they travel with the speed of light. Maxwell merged Gauss’s law, Fara- day’s law, Ampere’s law and form the four maxwell equation to explain their relationship.

The Heinrich has operated an experiment to show the claims from Maxwell; currents are applied in the two ends of two conductive wire, and it forms spark between the two terminals. The spark result in an electromagnetic wave trav- eling through the air which can be captured by a receiver. The gap between the spark(transmitter) and the receiver could glow up a light bulb. His research shows electromagnetic waves can be acted as transformations in space.

The Maxwell equation acts as a cornerstone of electrical and optical technologies, which brings motivation to telecommunication and wireless communication. The research re- garding electromagnetic waves and light has led to the discovery of quantum and relativity. Scientist continues their studies in the propagation of particles comparing with the speed of light.

2 1.2 Gauss’ Law for Electricity

1.2 Gauss’ Law for Electricity

Maxwell’s first equation is based on Gauss’s law, which mainly describes the relationship between and electric field. Let’s start with introducing the ’s law and explaining the relationship between eletric and electric field. Given two charges q1 and q2 at position r1 and r2, the force F1 exerted on q1 is

1 q1q2 F1 = 3 (r1 − r2). 4πε0 |r1 − r2|

From Coulomb’s law, we know that two negative charges will repel each other, while a positive charge and a negative charge will attract each other. The electric force is pro- portional to the product of the quantity of the charges, and square inversely proportional to their distance. By Law, electric field E is defined as the electric force F per unit charge q, i.e. F E = . q It follows that electric field is a vector field. Without loss of generality we assume that the point charge is at the origin. Then the electric field can be expressed as: q r E = 3 . 4πε0 r

3 In the following theorem we prove that electric field E is divergence-free in R \{0}.

3 Theorem 1.2.1. ∇ · E = 0 in R \{0}.

1 Proof. Let krk = px2 + y2 + z2 and F = (x, y, z). r3 By the definition of divergence,

∂r−3x ∂r−3y ∂r−3z divF = + + ∂x ∂y ∂z x y z = r−3 + x(−3r−4)( ) + r−3 + y(−3r−4)( ) + r−3 + z(−3r−4)( ) r r r = 3r−3 − 3r−5(x2 + y2 + z2) = 3r−3 − 3r−3 = 0. r q So ∇ · E = div( F) = divF = 0. 4πε0 4πε0

3 1 Maxwell Equations

Let Ω be a domain which doesn’t contain the origin. From previous theorem and diver- gence theorem we can immediately know that Z I ∇ · EdV = E · da = 0. (1.2.1) Ω ∂Ω

On the other hand, since E is no longer divergence-free at the origin, when the charge is R contained in the bounded region Ω, we have to find another way to calculate Ω ∇·EdV .

Theorem 1.2.2. Suppose a charge q is placed at the origin, and Ω be any domain containing the origin with smooth boundary ∂Ω. Then

Z q E · da = , ∂Ω 0 where da is the surface element on ∂Ω, q is the electric charge in coulomb (C), and ε0 represents the in coulomb square per newton meter square (C2Nm−2).

Proof. Suppose B is the ball centered at origin with radius r,

Z Z Z q r · r Z q 1 E · da = E · da = 3 = 2 ∂Ω ∂B ∂B 4πε0 |r| |r| ∂B 4πε0 |r| q Z 1 q 1 Z = 2 da = 2 da 4πε0 ∂B R 4πε0 R ∂B q 1 q 1 2 q = 2 × Area of B = 2 (4πR ) = . 4πε0 R 4πε0 R ε0

Theorem 1.2.2 is the Gauss’s Law for a single charge. To generalize Theorem 1.2.2 to multiple charges case, we need to take advantage of the following law of :

Theorem 1.2.3. Electrostatic obey the Principle of Superposition.

The principle of superposition stated that when there exist two or more charge in a bounded region, it follows that the total electric field at any point is the vector sum of the individual electric field created by each charge. So we have,

Z P q Q E · da = = , ∂R ε0 ε0 where R is a bounded region and Q is the total charge in R.

4 1.2 Gauss’ Law for Electricity

Theorem 1.2.4. For any closed surface ∂Ω bounded with a region Ω which contains a total charge of Q, we have I Q E · da = . ∂Ω ε0

The point charges (electrons) are just microscopic concepts. When we are studying macroscopic problems, it is very inconvenient to study the behaviour of each electron. Instead, we usually consider the charge distribution as a charge ρ(r), which is defined as the amount of charge in a unit . Suppose we are considering a sphere bounded with region Ω. As the can be defined as the amount of charge in a unit volume, the total charge can be represent as; Z 4 Q = ρdV = ρ( πr3). ∂Ω 3 Merging with the electric field, we have; 1 4 E = ( πr3). ε0 3 Taking integration on both sides, Z 1 Z E · da = ρdV ε0 ∂Ω and Z ρ (∇ · E − )dV = 0. ∂Ω ε

So we can conclude, ρ ∇ · E = . ε0

When studying from physics aspect, the Gauss’s law actually illustrates the total electric flux flowing out of a closed surface, where the flux is defined by I Q ΦE = E · da = ∂Ω ε0

It follows immediately that the flux is equal to the total charge Q enclosed in Ω divided by the permittivity ε0. Electric flux can be described as the flow of electric field. Suppose there is a charge q and electric field E is generated by this charge. Since E is a vector field, we could draw the integral line of E, which is vividly called “electric line”. Electric flux represents the amount of electric line passing through an area, and the electric flux passes through a surface can represented as the multiplication of electric field and the area of the surface

5 1 Maxwell Equations projected perpendicular to the field. The Gauss’s Law tells us that the area integral of the electric field over any closed surface is equal to the net charge enclosed in the surface divided by the permittivity of space. The physicists defined electric flux density D by the amount of electric flux passing through a defined area which is perpendicular to the orientation of the electric flux. With this notation the Gauss’s Law could be reformulated as

Corollary 1.2.4.1. I Z Q = D · da = ρdV, ∂Ω Ω where Q is the total electric charge, in coulomb (C), D stands for electric flux density in per square meter (Cm−2), and ρ represents charge density, in coulombs per cubic meter (Cm−3). This is usually regarded as the integral form of Maxwell’s first equation.

The integral form of Maxwell’s first equation explains when charge is distributed over a volume, the integration of charge density ρ by volume is equal to total charge Q. The electric flux density D is equal to the electric flux divided by the enclosed area, and the relationship between electric field and electric flux density can also be expressed as electric field equal to electric flux density divided by permittivity, where permittivity is known as the degree of the medium permitting the electric flux density. The integration of electric flux density by the enclosed surface area is also equal to charge q.

With the help of divergence theorem we could get another form of Maxwell’s first equa- tion

Corollary 1.2.4.2. ∇ · D = ρ. Here D and ρ shares the same meaning with previous corollary. This is regarded as the differential form of Maxwell’s first equation.

The differential form of Maxwell’s first equation says the divergence of electric flux den- sity is equal to the volume charge density, which means the more volume charge density, the more spreading of electric field.

6 1.2 Gauss’ Law for Electricity

Remark. Before proceeding to the next part, let’s briefly discuss the conservative nature of the electric field. A vector field is said to be conservative in a domain D if and only if there exist a scalar field φ, such that the vector field is equal to the gradient of D.

Theorem 1.2.5. Electric field is a conservative vector field.

q r − s Proof. We know that E(r) = 3 . Define 4πε0 |r − s| q 1 φ(r) = − . 4πε0 |r − s| By direct calculation we could verify that ∇φ = E.

The following theorem shows an important property of conservative vector field.

Theorem 1.2.6. The line integral of conservative vector field only depends on the start- ing point and ending, and is independent of choice of the trajectory.

Proof. Given a closed curve C = r(t) = x(t)i + y(t)j + z(t)k, a ≤ t ≤ b

∂F ∂F ∂F df ∂f ∂x ∂f ∂y ∂f ∂z f(x, y) = ∇F = i + j + k and = + + ∂x ∂y ∂z dt ∂x ∂t ∂y ∂t ∂z ∂t

Z Z b ∂F ∂x ∂F ∂y ∂f ∂z f · dr = ( + + )dt C a ∂x ∂t ∂y ∂y ∂z ∂t Z b dF b = dt = F (t)|a a dt = F (b) − F (a)

Theorem 1.2.7. By the nature of conservative vector field, the line integral along closed curve of conservative vector field is always zero.

∂F ∂F ∂F Proof. Given f(x, y) = ∇F = i + j + k, where f is a conservative vector field. ∂x ∂y ∂z Suppose a closed curve C is form by C1 and C2, where C1 and C2 share the same starting and ending point. Since F is conservative, the line integral of F is irrelevant with trajectory, but only depends on the starting point and ending point. Therefore we have Z Z Z Z f · dr = f · dr and f · dr − f · dr = 0. C1 C2 C1 C2

7 1 Maxwell Equations

Since Z Z f · dr = − f · dr, C2 −C2 it follows that Z Z Z Z f · dr + f · dr = f · dr − f · dr = 0 C1 C2 C1 C2

From physics aspect we can have another way to explain the previous result. A force is said to be conservative if the work done of moving a particle from one point to another point is based on the start and endpoint but not the travelling path. The work done on charge q in electric field is also followed by independent path, so the electric field also have the conservative nature.

Consider two point A and B in the electric field, a test charge q moves from point A to B along a closed path. Such movement can be expressed in line integral:

Z B Z A Z B Z B E · dl + E · dl = E · dl − E · dl = 0 A B A A

The divergence theorem stated that when R is a bounded region in 3-space with boundary surface ∂R. Given that f is a vector field, n is the unit normal vector pointing outward to ∂R, the relationship between area a and volume v integration of f can be expressed in the following equation: Z Z ∇ · fdv = f · nda R ∂R

1.3 Gauss’ Law for Magnetism

Maxwell’s second equation is also developed from Gauss’s law, which described the re- lationship between magnetic field and charge.

From Section 1.1, the Coulomb’s law stated that the electric charge produce electric field. When the charge is moving, it forms magnetic field, which means the electric currents I produce magnetic field B. Microscopically, when there is a charge q at the origin and moving with velocity v, Biot-Savart law tells us that it generates a magnetic field µ q v × r B = 0 . 4π r3

8 1.3 Gauss’ Law for Magnetism

When the charge is going around along a closed circuit C, by applying the Biot-Savart law, the resulting magnetic field can be shown as Z µ0 JdV × r B = 3 , 4π C r where J is the , µ0 is the permeability of free space.

The magnetic flux ΦM is defined as the amount of magnetic field lines passing through a specific area a, that is

I P magnetic pole Φ = B · da = , M k where k is a constant.

3 Theorem 1.3.1. ∇ · B = 0 in R \{0}.

µ Z JdV × r Proof. By Biot-Savart law, B = 0 , 4π r3 µ Z r ∇ · B = 0 ∇ · (J × )dV. 4π r3 Since ∇ · (A × B) = B · (∇ × A) − A · (∇ × B).

µ Z r r ∇ · B = 0 ( · (∇ × J) − J · (∇ × ))dV, 4π r3 r3 r as ∇ × J = 0 and ∇ × = 0, r3 ∇ · B = 0.

Theorem 1.3.2. Suppose a charge q is placed at the origin, and ω be any domain con- taining the origin with smooth boundary ∂ω. Then,

∇ · B = 0, where B is the magnetic field, in (T).

Proof. Suppose is a sphere centered at origin with radius r, I Z µ0q v × r r B · da = ( 3 ) · ( ) = 0. ∂R 4π ∂R r r

9 1 Maxwell Equations

By the divergence theorem, we can conclude that, Z ∇ · BdV = 0 R and ∇ · B = 0,

Corollary 1.3.2.1. I B · da = 0, where B is the electric field, in Tesla(T). This is usually regarded as the integral form of Maxwell’s second equation.

Corollary 1.3.2.2. ∇ · B = 0, where B is the electric field, in Tesla(T). This is named as the differential form of Maxwell’s second equation.

To conclude, according to Gauss’s law, the magnetic flux flowing out of any closed sur- face is zero. We know the magnetic sources are always dipole sources, since a single north pole or single south pole never exists. To compare with electric field, a north pole could be regarded as a “positive charge” and the south pole can be considered as a “negative charge”, and therefore in a closed domain containing the north pole and the south pole, the flux of magnetic field lines coming in to the south pole and going out from the north pole cancel out each other. As for magnetic dipole, in any closed surface, the amount of magnetic flux oriented inward toward the south pole will be same as the amount of flux moving outward from the north pole. So the net magnetic flux will always be zero.

1.4 Faraday’s Law of Induction

Maxwell’s third equation is developed from Faraday’s law, which discusses the relation- ship between induced electric field and the changing magnetic flux. Faraday’s law of induction illustrate a changing magnetic field will induce an electro- motive force (emf) in a coil of wire. Suppose there is a coil of wire, and we have a moving inside the coil, then a current will flow. If the magnetic field is steady, no current will be formed. The induced current arises because of the change of magnetic field. The faster the magnetic field is changing, the greater the current will generate. (Emf) is the potential difference causing the current to flow, so we

10 1.4 Faraday’s Law of Induction can say the Emf is proportional to the rate of change of magnetic field over time. Emf is also proportional to the area of the surface which is surrounded by the current loop. Now we can define the magnetic flux ΦM to be the amount of magnetic field lines that pass through a closed surface Ω, Z ΦM = B · da. Ω

If we want to know the relationship between the magnetic flux ΦM and magnetic field B in the changing time, then by taking time derivative on both side we get

dΦ Z dB M = · da. (1.4.1) dt Ω dt As emf is the force of a changing magnetic field within a specific area, so we have

dΦ Emf = − M , (1.4.2) dt where the negative sign is because of the Lenz’s law, as the direction of the induced is opposite to the changing magnetic field.

In other words, electromotive force (Emf) is the potential energy (PE) acting on a charge to move from one point to another point, namely potential energy by unit charge. The emf is the work done to bring a unit charge q for a distance to a destination. Soppose we are moving the charge along a curve C Z PE = workdone = F · dl, C

Where F is force and dl is the distance of the line segment in curve C.

From Coulomb’s law, we know

Qq F = 3 r, 4πε0r When there are two charge Q and q. Inserting above result to the potential energy (PE), we have

Z Qq r PE = workdone = 3 · dl. C 4πε0 |r| Merging with the emf, emf is defined as workdone per unit charge

PE Z Qq r Emf = = 3 · dl. q C 4πε0 |r|

11 1 Maxwell Equations

By Maxwell’s first equation

Q r E = 3 , 4πε0 |r|

We can then conclude, Z Emf = E · dl. C where E is the electric field, emf is in SI unit per meter (V/m).

Theorem 1.4.1. For the integral form, by combining the results from Equation (1.4.1), (1.4.2) we get

I Z ∂B E · dl = − · da L A ∂t where E is the elctric field, in newtons per coulomb (NC−1), B stand for electric field, in Tesla(T). This is named as the integral form of Maxwell third equation.

Theorem 1.4.2. ∂B ∇ × E = − ∂t where E is the electric field, in newtons per coulomb (NC−1) and B is the electric field, in Tesla(T). This is regarded as the differential form of Maxwell’s third equation.

To conclude, the general form of the law of induction is: I dΦ ∂ Z Emf = E · dl = − M = − B · da L dt ∂t A The left-hand side of the equation indicates the line integration of electric field E over a closed line L which bounds the surface area A. The line integral sum up all the parts of the electric field passing through the line L and it measures how much electric field rotate along the line L. It is equal to the right-hand side, i.e. the surface integral of magnetic field B over a surface A, which means the magnetic flux ΦM through a surface A is differential respond to time t, it illustrate how much magnetic flux change due to the change of time. The larger the change of magnetic flux, the larger the rotation among the electric field. The minus sign denoted by lenz’s law shows the direction of rotation. To summerize, the Maxwell-Faraday equation states the change of magnetic flux in a closed surface A which creates emf, and the tangential part of electric field E along boundary L of A also creates the same amount of emf.

12 1.5 Ampere’s Law

Now, let’s consider the special case if the magnetic field does not change over time. ∂B The magnetic field is time-independent ( ∂t = 0), so the right side of the maxwell’s equation will be zero, which means the emf along the closed line is zero, there is no rotating electric field. This can be shown by the following equation:

∇ × E = 0.

1.5 Ampere’s Law

The fourth Maxwell’s equation is developed from the Ampere’s law, which discusses the relationship between magnetic field, electric field and current.

Let’s first introduce the Ampere’s law. Ampere’s law says that a current passing through a coil of wire will induces a magnetic field around the path surrounding the wire. Con- sider a current which is flowing in a coil of wire on xy-plane counterclockwisely. Around the wire we will find a magnetic field B. The direction of B is along the positive z-axis, and the magnitude of B is proportional to the current I, which means the greater the current is, the greater magnetic field divided by r is, where r is the distance from the wire to the magnetic field. Magnetic field can be produced by moving charges. Suppose there is a electric current flowing along the curve C, by the Biot-Savart law,

µ I ds × (r − s) dB = 0 , 4π |r − s|3 where B is the magnetic field, µ0 is the permeability of free space, I is the current. Taking integration on both side, we have I µ0I ds × (r − s) B = 3 . 4π C |r − s| Let’s define a vector field A, µ I I ds A = 0 . 4π C |r − s| We have known from previous sections that

1 r − s ∇( ) = − , |r − s| |r − s|3 therefore by taking the of A we get I µ0I r − s ∇ × A = − 3 × ds = B(r), 4π C |r − s| which implies that A is a vector potential for B.

13 1 Maxwell Equations

Now we consider another case. Suppose there is a current in a straight wire along the z-axis, the magnetic field B will then be produced in the surrounding of the straight wire due to the current which is along z-axis. The current is produced in ds, where the magnetic field B is normal to the plane containing r. By applying Bivot-Savart law, we can calculate the magnetic field B with radius a from the z-axis as

µ I sin θdb µ I adb dB = 0 = 0 , 4π a2 + (b − z)2 4π (a2 + (b − z)2)3/2 where b is the z-intercept. Then, we let b − z = a tan φ and

µ Ia Z ∞ db B = 0 2 2 3/2 4π −∞ (a + (b − z) ) π µ I Z 2 µ I = 0 cos φdφ = 0 . 4π π 2πa − 2

Now suppose C is a circle on xy-plane centered at origin with radius a. Similar to the proof of Faraday’s law, we can break the C into small segments with length dl, and we have I µ0I B · dl = B(2πa) = (2πr) = µ0I. C 2πa

As current can be defined as the current density within a specific area a, the relationship of current and current density can be shown as; Z I = J · da, where I is the current, J is the current density.

Lemma 1.5.1. The differential form of Ampere’s Law can be written as ∇ × B = µ0J.

Proof. I Z B · dl = µ0I = µ0 J · da. C So we can conclude,

∇ × B = µ0J.

14 1.5 Ampere’s Law

We know the electric field and current density both depend on time, so we need modifi- cation on it. Recall from the integral form of Maxwell’s first equation, we know the relationship be- tween the total charge and the charge density. Then, we want to know the rate of change of total charge in a region R;

∂ ZZZ ZZZ ∂ρ ρdV = dV, ∂t R R ∂t where ρ is the charge density. By conservation of charge, the equation can be express in form of current density, I ZZZ (−J) · Ndaˆ = − ∇ · JdV. R Then we have

ZZZ ∂ρ ( + ∇ · J)dV = 0 R ∂t and

∂ρ = −∇ · J. ∂t In Maxwell’s first equation, we know ∇ · E = ρ , combining with the current density: ε0 ∂ρ ∂E = −∇ · J = ε ∇ · ∂t 0 ∂t ∂E So ∇ · (J + µ0ε0 ∂t ) = 0. With the time related term added into the equation, the Maxwell-Ampere equation be- comes:

∂E ∇ × B = µ J + µ ε . 0 0 0 ∂t

The Maxwell-Ampere equation can be expressed in two forms;

Corollary 1.5.1.1. I ∂ I B · dl = µ I + µ ε E · da 0 0 0 ∂t where B is the electric field, in Tesla(T), I stand for current, in ampere (A), µ0 is −2 −2 the pemeability of free space, in SI unit mkgs A , ε0 is the permittivity, in coulomb square per newton meter square (C2Nm−2) and E stand for elctric field, in newtons per coulomb (NC−1). This is usually regarded as the integral form of Maxwell’s fourth equation.

15 1 Maxwell Equations

Corollary 1.5.1.2. ∂E ∇ × B = µ J + µ ε 0 0 0 ∂t Where B is the electric field, in Tesla(T), J stand for current density, in ampere per square meter (Am−2) and E is the elctric field, in newtons per coulomb (NC−1). This is named as the differential form of Maxwell’s fourth equation.

To conclude, the general form of the equation is:

I ∂ Z B · dl = µ0I + µ0ε0 E · da L ∂t A

The left side of the equation implies the line integral of the magnetic field B along the closed line L. On the other hand, we know when charge flows along a conductor, it generate current I, the area integral of electric field E is the electric flux through a sur- face, in addition, a time derivative is ahead of the electric flux, it shows the change of eleectric flux. There are two term sum up in the right hand side, one of which is the contribution of current and the other one is contributed by the electric flux over time. As a result, the Maxwell’s fourth equation implies that a rotating magnetic field can be generated by the electric current through the surface and by the changing of electric field.

16 2 Special Cases and Boundary Conditions

After deriving the Maxwell’s equations in general form, we are interested in how the Maxwell’s equation can be expressed in some special cases.

2.1 Electromagnetic wave in Vacuum

Let’s consider the Maxwell’s equations in vacuum. In vacuum, there is no charge and current, so the charge density ρ and current density J can be eliminated. Which means ρ = 0 and J = 0. The Maxwell’s equations in vacuum can be shown as:

∇ · E = 0, (2.1.1) ∇ · B = 0, (2.1.2) ∂B ∇ × E = − , (2.1.3) ∂t ∂E ∇ × B = µ ε , (2.1.4) 0 0 ∂t where E is the electric field and B is the magnetic field.

Taking curl on equation 2.1.3, we have the wave equation for electric field;

∂B ∂ ∂2E ∇ × (∇ × E) = −∇ × = − (∇ × B) = −µ ε . ∂t ∂t 0 0 ∂t2 Using the identity ∇ × (∇ × F) = ∇(∇ · F) − ∇2F and the fact that ∇ · E = 0, we can conclude,

1 ∂2E ∇2E − = 0, (2.1.5) c2 ∂t2 1 where c = √ = 3 × 108m/s, which is the speed of light. This equation follows the µ0ε0 standard form of wave equation.

Taking curl on equation 2.1.4, we can obtain the wave equation for magnetic field;

∂E ∂ ∂2B ∇ × (∇ × B) = −∇ × (µ ε ) = − µ ε (∇ × E) = −µ ε . 0 0 ∂t ∂t 0 0 0 0 ∂t2

17 2 Special Cases and Boundary Conditions

Using the identity ∇ × (∇ × F) = ∇(∇ · F) − ∇2F and the fact that ∇ · B = 0, we can conclude,

1 ∂2B ∇2B − = 0, (2.1.6) c2 ∂t2 1 where c = √ = 3 × 108m/s, which is the speed of light. This equation follows µ0ε0 the standard form of wave equation. To summarize, the Maxwell equation in vacuum is equivalent to the following equation system of E and B

 1 ∂2E ∇2E − = 0,  c2 ∂t2 1 ∂2B ∇2B − = 0.  c2 ∂t2

2.2 Time Harmonic Field

To introduce time harmonic field, let’s start with considering transverse polarized waves. We assume that E is only propagating in xz-plane along the positive direction of z-axis (which also means that the magnetic field B is propagating only in yz-plane). By seperation of variables, we assume E is in the following form

E =xE ˆ x. ∂E Since ∇ · E = 0 we get x = 0. ∂x

We assume E =xE ˆ x(z, t) Merging with equation 2.1.5, we have

∂2E (z, t) 1 ∂2E (z, t) x = x , ∂z2 c2 ∂t2 and

Ex(z, t) = g(z ± t), where function g is depends on coordinate-z with time t which satisfy the above equation.

To specify, let’s consider the plane wave of sinusoid, we have

2π 2πc 2π E =xE ˆ (z − ct) =xE ˆ cos( (z − ct)) =xE ˆ cos( t − z), x 0 λ 0 λ λ where E is electric field, pointing in x-direction, and having wavelength λ, frequency = c , moving to the z-direction. λ

18 2.3 Boundary Conditions

2πc 2π Let ω = λ and k = λ , we get

E =xE ˆ 0 cos(ωt − kz). (2.2.1) Let’s also consider the magnetic field, from equation 2.1.3;

∂B ∇ × E = − . ∂t Combining with the result we got above regarding the sinusoids, we have

∂B = −∇ × E = −ykEˆ sin(ωt − kz), ∂t 0 Taking integration, we get

k B =y ˆ E (cos(ωt − kz)). (2.2.2) ω 0 From equation 2.2.1, we can write it in more general form:

E(r, t) =nE ˆ 0 cos(ωt − k · r), where the electric field E has position r, in directionn ˆ and going to some direction in 3-space. From this result, we can get

i(ωt−k·r E(r, t) = Re[ˆnE0e )] −ik·r iωt = Re[ˆnE0e e ] = Re[E(r)eiωt],

−ik·r where E(r) =nE ˆ 0e . This is the derivation of time harmonic electric field. Notice that E(r) is complex while the true electric field is real. The field can also be regarded as monochromatic field.

Plug in this result to equation 2.1.5, we can get the Helmholtz equation;

∇2E(r) + k2E(r) = 0, ω with k = . By the same reasoning, we can get a similar equation for magnetic field c ∇2B(r) + k2B(r) = 0.

2.3 Boundary Conditions

When the electromagnetic wave is propagating through different medium, we need to derive corresponding boundary conditions to illustrate the propagation behaviour on the

19 2 Special Cases and Boundary Conditions boundary. To be specific, the boundary conditions study the situation in a plane which lies between two medium, where we are interesting in the behaviour of electric field E, magnetic field B, electric flux density D and magnetic flux density H. Suppose we have a cylinder V , with thickness σ, and ∆A be the area of the and bottom surface, there exist a boundary cuts the cylinder which seperate two homogenous medium. Let’s first consider the normal component of the electric field.

By Gauss’s theorom, ZZZ ZZ ∇ · DdV = D · dS V ∂V ZZ ZZ ZZ = Dtop · dS + Dbottom · dS + Dside · dS. top bottom side RR Let σ tends to zero, side D · dS = 0. We then assume ∆A to be very small, D on the top and bottom surface would approx- imately regarded as constants. So ZZ Dtop · dS = (Dtop · n) · ∆A, top ZZ Dbottom · dS = (Dbottom · (−n)) · ∆A, botttom and ZZZ ∇ · DdV = (Dtop · n) · ∆A − (Dbottom · n) · ∆A V = ∆A[(Dtop · n) − (Dbottom · n)].

From Maxwell’s first equation, we know

∇ · D = ρ and ZZZ ZZZ ∇ · DdV = ρdV V V From physics we know that free charge density can only appear on the medium bound- ary, so ZZ ρdS = ρ · ∆A S and ∆A[(Dtop · n) − (Dbottom · n)] = ρ · ∆A. We can conclude that

20 2.3 Boundary Conditions

Theorem 2.3.1. n · [Dtop − Dbottom] = ρ, where D is the electric flux density and ρ is the charge density. This implies that the normal part of D differs by the surface charge density across the boundary.

We then consider the tangential component of the electric field. Assume that there is a rectangular path across the boundary. The rectangle can be named as rectangle abcd, the length are represented as C1(ab) and C3(cd), and the width are C2(bd) and C4(ac).

From Maxwell’s third equation, we know

∂B ∇ × E = − . ∂t Applying stoke’s theorem, we can get

Z Z Z ∂B E · dl = (∇ × E) · dS = − · dS. C1+C2+C3+C4 R ∂t Let the width of the rectangular path tends to zero, so C2 and C4 tend to zero and ∂B − = 0. Then we have ∂t Z E · dl = 0, C1+C3 Z Z E · dl = − E · dl, C1 C3 Z b Z d Etop · dl = Ebottom · dl, a c and Z b Etop · dl = (tangential part of Etop) · ∆l a = (tangential part of Ebottom) · ∆l, where the length of C1 and C3 are represent by ∆l. Therefore, we can conclude

Theorem 2.3.2. n × [Etop − Ebottom] = 0, where E is the electric field. This implies that the tangential part of E is continuous across the boundary.

By the same reasoning, we can apply Maxwell’s second and fourth equation and get similar results regarding the normal part and the tangential part of the magnetic field as follows

21 2 Special Cases and Boundary Conditions

Theorem 2.3.3. n · [Btop − Bbottom] = 0, where B is the magnetic field. This implies that the normal part of B is continuous across the boundary.

Theorem 2.3.4. n × [Htop − Hbottom] = J, where H is the magnetic flux density and J is the current density. This implies that the tangential part of H differs by the surface current density across the boundary.

22 3 Hyperbolic metamaterial

Metamaterial is the name of a vast category of man-made material with properties that never occur on natural materials [7]. It consists of assemblies of multiple repeatedly aligned unit structures made of metal or plastics. The size of these unit structures are smaller than the wavelength of the phenomena they affect. In acoustic metamaterial, the unit size is usually of centimeter order, while for optic metamaterial the size varies from millimeter order to nanometer order [11, 12, 13]. The most significant physics property of metamaterial is a sharp change in its physical characteristics at some specific frequencies, which are usually called “resonant frequencies”. Take optic metamaterial as an example. In normal case, the small inclusion array wouldn’t change the permittivity  and permeability µ too much, and by default the background  and µ are both positive. Nevertheless at resonant frequencies, the permittivity  or the permeability µ, or even both of them could become negative.

The physical nature of metamaterial does not solely depend on the material of its unit structure. To a great extent it relies on the shape, size, alignment and orientation of its unit structure. By adjusting those properties, we could expect a remarkable change of corresponding metamaterial in the absorption, enhancement or refraction to incident wave, of which we could take advantage to develop new materials possessing specific physical properties.

The study on metamaterial originated from the end of [1]. In 1904, and both mentioned negative phase velocity accompanied by anti- parallel group velocity [16], but researchers back then thought this phenomenon is not likely to occur in practical. V.G. Veselago’s 1967 paper is considered the theoretical work that began metamaterial research [16], and with the advance of material manufacturing technology, the fine structures depicted in papers were produced in reality by the end of 20th century, and now metamaterial owns a broad potential application field, including optical filters, medical device, remote application, sensor detection, infrastructure mon- itoring, smart solar panel management, high-frequency battlefield communication and superlens [15].

Now let’s discuss the simplest metamaterial for electromagnetic wave. Suppose the incident electromagnetic wave is transverse electric polarized, which means we only have to take care of the magnetic field B, and the equation dominating the propagation model is Helmholtz equation. The building units of the metamaterial are exactly , whose sizes are of order δ  1 and with electric permittivity εc and magnetic permeability µc. They are both functions of the frequency of the incident wave. We need

23 3 Hyperbolic metamaterial

to further assume here that Im εc > 0, Re µc < 0, Im µc > 0. The µc must be negative since otherwise we couldn’t expect any resonance in the whole frequency band, and the positive imaginary part of permittivity and permeability refers to the dissipation nature of the particles. Let εm, µm denote the permittivity and permeability of free space, correspondingly. The xy-plane is total reflective, i.e. the electric field satisfies Dirichlet boundary condition. Define √ √ km = ω εmµm, kc = ω εcµc.

Here we assume that εm, µm > 0 and km is of order 1. The metamaterial is constructed as shown in Figure 3.1, where the unit particle D is periodically aligned along the x-axis with period and interval both of order δ above the 2 2 boundary y = 0 of the half-space R+ := {(x, y) ∈ R , y > 0}. We denote D as the whole 2 set of periodically arranged particles and Ω := R+ \ D.

Figure 3.1: Example of metamaterial

From our discussion in previous sections, the whole system could be formulated as  1  ∇ · ∇u + ω2 u = 0 in 2 \ ∂D,  D R+  µD   u+ − u− = 0 on ∂D, (3.0.1) 1 ∂u 1 ∂u  − = 0 on ∂D, µ ∂ν µ ∂ν  m + c −  2  u = 0 on ∂R+ = {(x1, 0), x1 ∈ R}, where εD = εmχ(Ω) + εcχ(D), µD = µmχ(Ω) + µcχ(D), and ∂/∂ν denotes the outward normal derivative on ∂D. Recently, a new form of metamaterial, which is called hyperbolic metamaterial, is draw- ing increasing attention from researchers in material science, theoretical physics and

24 mathematics. The most striking difference between hyperbolic metamaterial and other materials is that hyperbolic metamaterial is highly anisotropic with hyperbolic disper- sion determined by their effective electric/magnetic tensors. Hyperbolic metamaterial could be regarded as a representation of the ultra-anisotropic limit of traditional uni- axial crystals. In hyperbolic metamaterial, one of the principal components of either permittivity or permeability must bear the opposite sign against the other two princi- pal components [9]. When considering transverse magnetic polarized wave, if material permittivity tensor is supposed to be in the form of   x 0 0 ˆ =  0 y 0  0 0 z and wave vector corresponds to incident electromagnetic wave to be k = (kx, ky, kz), then from dispersion relation it follows that

k2 k2 k2 ω 2 x + y + z = x y z c where ω and c stands for frequency of incident wave and light speed, correspondingly. When x, y > 0 and z < 0, dispersion relation becomes

k2 k2 k2 ω 2 x + y − z = |x| |y| |z| c and the resulting isofrequency surface appears as hyperbolic surface, leading to the name of hyperbolic metasurface. When the incident electromagnetic wave whose wave-vector magnitude is large, impinges on the hyperbolic metasurface, we could expect the observe some very important phe- nomenons. In vacuum or other isotropic medium, the isofrequency contour is simply a circle or an elliptic curve, which implies that wave with large wave-vector magnitude will only be an evanescent wave. Nevertheless, in hyperbolic media the hyperbolicity of the isofrequency contour allows the propagation of wave with infinite large wave-vectors in the idealistic limit [4, 5], and therefore evanescent wave does not exist in such medium. This exotic property gives rise to a wide range of possibilities of device applications with hyperbolic metamaterial [10].

25 4 Conclusion

In this report, we have introduced Maxwell’s equation merging with the concepts of Gauss’s law, Faraday’s law, and Ampere’s law. We have provided a full derivation for the four Maxwell’s equations in mathematical aspects, with explanations regarding the relationship between electric field and magnetic field. Moreover, we have investigated the simplified form of Maxwell’s equations in vacuum and time-harmonic fields. We also discovered the expression of Maxwell’s equations under boundary conditions. In the last part of the report we briefly introduced metamaterial by studying the equation of the simplest model. Hyperbolic metamaterial is also presented at the very end, but unfortunately due to the limit of project length we could not carry out an in-depth research on the mathematical principle of this interesting material.

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