Maxwell Equations and Hyperbolic Metamaterial by KWAN NGA WONG 17220955 A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science (Honours) in 2021 at Hong Kong Baptist University 29 December 2020 Acknowledgement The thesis was my original work 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 Mathematics Hong Kong Baptist University December 29, 2020 ii Abstract In this project, we are going to introduce Maxwell's equation which merged the concepts of Gauss's law, Faraday's law, and Ampere'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 Electricity . .3 1.3 Gauss' Law for Magnetism . .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 wave in Vacuum . 16 2.2 Time Harmonic Field . 17 2.3 Boundary Conditions . 18 3 Hyperbolic metamaterial 22 4 Conclusion 25 Bibliography 26 1 1 Maxwell Equations 1.1 Introduction Electromagnetic waves 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 light and electromagnetic waves[14, 6, 8]. In the 17th century, Issac Newton 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 polarization 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 speed of light. 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 scientist Heinrich Hertz 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 energy 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 physics 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 electric charge and electric field. Let's start with introducing the Coulomb's law and explaining the relationship between eletric force 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): 4π"0 jr1 − r2j 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 Lorentz Force 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 : 4π"0 r 3 In the following theorem we prove that electric field E is divergence-free in R nf0g. 3 Theorem 1.2.1. r · E = 0 in R nf0g. 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 r · E = div( F) = divF = 0: 4π"0 4π"0 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 r · 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 r·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 "0 represents the permittivity 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 4π"0 jrj jrj @B 4π"0 jrj q Z 1 q 1 Z = 2 da = 2 da 4π"0 @B R 4π"0 R @B q 1 q 1 2 q = 2 × Area of B = 2 (4πR ) = : 4π"0 R 4π"0 R "0 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 electrostatics: Theorem 1.2.3. Electrostatic forces 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 "0 "0 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 "0 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 density ρ(r), which is defined as the amount of charge in a unit volume. Suppose we are considering a sphere bounded with region Ω. As the charge density 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): "0 3 Taking integration on both sides, Z 1 Z E · da = ρdV "0 @Ω and Z ρ (r · E − )dV = 0: @Ω " So we can conclude, ρ r · E = : "0 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 "0 It follows immediately that the flux is equal to the total charge Q enclosed in Ω divided by the permittivity "0. 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.