Transcript Chapter 29
Chapter 29 Maxwell’s Equations and Electromagnetic Waves Maxwell’s Theory • Electricity and magnetism were originally thought to be unrelated • Maxwell’s theory showed a close relationship between all electric and magnetic phenomena and proved that electric and magnetic fields play symmetric roles in nature • Maxwell hypothesized that a changing electric field would produce a magnetic field • He calculated the speed of light – 3x108 m/s – and concluded that light and other electromagnetic waves consist of James Clerk Maxwell fluctuating electric and magnetic fields 1831-1879 Maxwell’s Theory • Stationary charges produce only electric fields • Charges in uniform motion (constant velocity) produce electric and magnetic fields • Charges that are accelerated produce electric and magnetic fields and electromagnetic waves • A changing magnetic field produces an electric field • A changing electric field produces a magnetic field • These fields are in phase and, at any point, they both reach their maximum value at the James Clerk Maxwell same time 1831-1879 Modifications to Ampère’s Law • Ampère’s Law is used to analyze magnetic fields created by currents • But this form is valid only if any electric fields present are constant in time • Applying Ampère’s law to a circuit with a changing current results in an ambiguity • The result depends on which surface is used to determine the encircled current. B ds 0 I Modifications to Ampère’s Law • Maxwell used this ambiguity, along with symmetry considerations, to conclude that a changing electric field, in addition to current, should be a source of magnetic field • Maxwell modified the equation to include time-varying electric fields and added another term, called the displacement current, Id • This showed that magnetic fields are produced both by conduction currents and by time-varying electric fields d E B d s 0 I 0 0 dt d E Id 0 dt Maxwell’s Equations • In his unified theory of electromagnetism, Maxwell showed that the fundamental laws are expressed in these four equations: B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • Gauss’ Law relates an electric field to the charge distribution that creates it • The total electric flux through any closed surface equals the net charge inside that surface divided by o B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • Gauss’ Law in magnetism: the net magnetic flux through a closed surface is zero • The number of magnetic field lines that enter a closed volume must equal the number that leave that volume • If this wasn’t true, there would be magnetic monopoles found in nature B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • Faraday’s Law of Induction describes the creation of an electric field by a time-varying magnetic field • The emf (the line integral of the electric field around any closed path) equals the rate of change of the magnetic flux through any surface bounded by that path B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • Ampère-Maxwell Law describes the creation of a magnetic field by a changing electric field and by electric current • The line integral of the magnetic field around any closed path is the sum of o times the net current through that path and oo times the rate of change of electric flux through any surface bounded by that path B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be found F qE qv B • Maxwell’s equations with the Lorentz Force Law completely describe all classical electromagnetic interactions B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Maxwell’s Equations • In empty space, q = 0 and I = 0 • The equations can be solved with wave-like solutions (electromagnetic waves), which are traveling at the speed of light • This result led Maxwell to predict that light waves were a form of electromagnetic radiation B dA 0 d E B d s 0 I 0 0 dt q E dA 0 d B E d s dt Electromagnetic Waves • From Maxwell’s equations applied to empty space, the following relationships can be found: E E 0 0 2 2 x t 2 2 B B 0 0 2 2 x t 2 2 • The simplest solutions to these partial differential equations are sinusoidal waves – electromagnetic waves: E E max cos( k x t ); B B max cos( k x t ) • The speed of the electromagnetic wave is: 1 E E max 8 v c 2 .9 9 7 9 2 1 0 m/s k B Bmax 0 0 Plane Electromagnetic Waves • The vectors for the electric and magnetic fields in an em wave have a specific spacetime behavior consistent with Maxwell’s equations • Assume an em wave that travels in the x direction • We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only • The electric field is assumed to be in the y direction and the magnetic field in the z direction Plane Electromagnetic Waves • The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation • Thus, electromagnetic waves are transverse waves • Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves Poynting Vector • Electromagnetic waves carry energy John Henry Poynting can 1852 – 1914 • As they propagate through space, they transfer that energy to objects in their path • The rate of flow of energy in an em wave is described by a vector, S, called the Poynting vector defined as: S 1 E B μo • Its direction is the direction of propagation and its magnitude varies in time • The SI units: J/(s.m2) = W/m2 • Those are units of power per unit area Poynting Vector • Energy carried by em waves is shared equally by the electric and magnetic fields • The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles • When the average is taken, the time average of cos2(kx ωt) = ½ is involved I S avg 2 2 E max Bmax E max c Bmax 2 μo 2 μo c 2 μo Chapter 29 Problem 29 What would be the average intensity of a laser beam so strong that its electric field produced dielectric breakdown of air (which requires Ep = 3 MV/m)? Polarization of Light • An unpolarized wave: each atom produces a wave with its own orientation of E, so all directions of the electric field vector are equally possible and lie in a plane perpendicular to the direction of propagation • A wave is said to be linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point • Polarization can be obtained from an unpolarized beam by selective absorption, reflection, or scattering Polarization by Selective Absorption • The most common technique for polarizing light • Uses a material that transmits waves whose electric field vectors in the plane are parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction Polarization by Selective Absorption • The intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as S = So cos2 θ, where So is the intensity of the polarized wave incident on the analyzer • This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other Étienne-Louis Malus 1775 – 1812 Chapter 29 Problem 40 A polarizer blocks 75% of a polarized light beam. What’s the angle between the beam’s polarization and the polarizer’s axis? Electromagnetic Waves Produced by an Antenna • Neither stationary charges nor steady currents can produce electromagnetic waves • The fundamental mechanism responsible for this radiation: when a charged particle undergoes an acceleration, it must radiate energy in the form of electromagnetic waves • Electromagnetic waves are radiated by any circuit carrying alternating current • An alternating voltage applied to the wires of an antenna forces the electric charge in the antenna to oscillate Electromagnetic Waves Produced by an Antenna • Half-wave antenna: two rods are connected to an ac source, charges oscillate between the rods (a) • As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b) • The charges and field reverse (c) and the oscillations continue (d) Electromagnetic Waves Produced by an Antenna • Because the oscillating charges in the rod produce a current, there is also a magnetic field generated • As the current changes, the magnetic field spreads out from the antenna • The magnetic field lines form concentric circles around the antenna and are perpendicular to the electric field lines at all points • The antenna can be approximated by an oscillating electric dipole The Spectrum of EM Waves • Types of electromagnetic waves are distinguished by their frequencies (wavelengths): c = ƒ λ • There is no sharp division between one kind of em wave and the next – note the overlap between types of waves The Spectrum of EM Waves • Radio waves are used in radio and television communication systems • Microwaves (1 mm to 30 cm) are well suited for radar systems + microwave ovens are an application • Infrared waves are produced by hot objects and molecules and are readily absorbed by most materials The Spectrum of EM Waves • Visible light (a small range of the spectrum from 400 nm to 700 nm) – part of the spectrum detected by the human eye • Ultraviolet light (400 nm to 0.6 nm): Sun is an important source of uv light, however most uv light from the sun is absorbed in the stratosphere by ozone The Spectrum of EM Waves • X-rays – most common source is acceleration of high-energy electrons striking a metal target, also used as a diagnostic tool in medicine • Gamma rays: emitted by radioactive nuclei, are highly penetrating and cause serious damage when absorbed by living tissue Answers to Even Numbered Problems Chapter 29: Problem 14 3.9 μA Answers to Even Numbered Problems Chapter 29: Problem 22 (a) 3 m (b) 6 cm (c) 500 nm (d) 3 Å Answers to Even Numbered Problems Chapter 29: Problem 32 (a) 160 W/m2 (b) 350 V/m (c) 1.2 μT Answers to Even Numbered Problems Chapter 29: Problem 36 3.1 cm