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Physics 42200 Waves & Oscillations Lecture 25 – Introduction to Optics Spring 2015 Semester Matthew Jones Introduction to Optics • If we can identify any system that is described by the wave equation, then we know everything! 1 = • Except that… – – – – The geometry might be complicated Boundary conditions might not be simple Initial conditions might not be known The solution is the superposition of incident waves and all possible reflected waves • So we develop new ways of looking at this problem and call it optics. Electromagnetism • Geometric optics overlooks the wave nature of light. – Light inconsistent with longitudinal waves in an ethereal medium – Still an excellent approximation when feature sizes are large compared with the wavelength of light • But geometric optics could not explain – Polarization – Diffraction – Interference • A unified picture was provided by Maxwell c. 1864 Forces on Charges • Coulomb’s law of electrostatic force: $# Charles-Augustin de Coulomb (1736 - 1806) $ • The magnitude of the attractive/repulsive force is where = ̂ 1 = = 8.99 × 10 ∙ ∙ 4 and therefore ! = 8.85 × 10!# ∙ !# ∙ ! (This constant is called the “permittivity of free space”) Electric Field • An electric charge changes the properties of the space around it. – It is the source of an “electric field”. – It could be defined as the “force per unit charge”: # = 5% %= ̂ &'() – Quantum field theory provides a deeper description… • Gauss’s Law: 1 $012034 *, + ∙ %-. = / Electric “flux” through surface S Magnetic Field • Moving charges (ie, electric current) produce a magnetic field: 7 8-ℓ × ̂ * 6= 4 A moving charge in a magnetic field experiences a force: = 5 × 6 Lorentz force law: = 5% + 5 × 6 Gauss’s Law for Magnetism • Electric charges produce electric fields: $012034 , ∙ %-. = *+ / • But there are no “magnetic charges”: 2 , ∙ 6-. = *+ / 0 Magnetic “flux” through surface S Ampere’s Law • An electric current produces a magnetic field that curls around it: ; ; (Close, but not quite the whole story) Faraday’s Law of Magnetic Induction • Magnetic flux: <= = >/ 6 ∙ +? -. • Faraday observed that a changing magnetic flux through a wire loop induced a current – It transferred energy to the charge carriers in the wire -<= ℇ = − - B % ∙ -ℓ = − * +? ∙ 6-. - / ; 3 The Problem with Ampere’s Law Current through C# is 8 Current through C is 0! The current through C is zero, but the electric flux is not zero. The electric flux changes as charge flows onto the capacitor. Maxwell’s Displacement Current • We can think of the changing electric flux through C as if it were a current: -<4 83 = = * % ∙ +? -. - - / B D ∙ Eℓ = GH I + IE F E , EQ = GH J + GH KH * M ∙ N EL OP 4 Maxwell’s Equations (1864) RSNTSEU , ∙ M EQ = B N KH O , ∙ D EQ = H B N O EVW B M ∙ Eℓ = − EL F EVU B D ∙ Eℓ = GH I + GH XH EL F 1 2 3 4 Maxwell’s Equations in Free Space In “free space” where there are no electric charges or sources of current, Maxwell’s equations are quite symmetric: , ∙ M EQ = H B N O , ∙ D EQ = H B N O EVW B M ∙ Eℓ = − EL F EVU B D ∙ Eℓ = GH XH EL F Maxwell’s Equations in Free Space EVW B M ∙ Eℓ = − EL F EVU B D ∙ Eℓ = GH XH EL F A changing magnetic flux induces an electric field. A changing electric flux induces a magnetic field. Will this process continue indefinitely? Light is an Electromagnetic Wave • Faraday’s Law: ∮; % ∙ -ℓ = 3Z[ − 3\ =− ]^_ ]\ ∆a∆b B % ∙ -ℓ = %c b Δa − %c (b# )Δa ; ≈ x ]hi ∆b∆a ]j Ex z1 y By Δa Ex Δb z2 z Light is an Electromagnetic Wave • Ampere’s law: ∮; 6 ∙ -ℓ = 3Zk 7 3\ = ]hi 7 ∆l∆b ]\ B 6 ∙ -ℓ = 6m b# Δl − 6m (b )Δl ; ≈− ]^_ ]j x ∆b∆l Ex ∆b z1 z2 y By By ∆l z Putting these together… 6m %c =− b 6m %c − = 7 b Differentiate the first with respect to b: 6m %c =− b b Differentiate the second with respect to : 6m %c − = 7 b nP M o nP Mo = GH KH P np nLP Velocity of Electromagnetic Waves l 1 l = a nP M o nP M o = GH KH P np nLP Speed of wave propagation is 1 = 7 1 = 4 × 10!q /. 8.854 × 10!# = P. sst × uHt v/w (speed of light) / ∙ Light is an Electromagnetic Wave • Maxwell showed that these equations contain the wave equation: 1 l l = a where l a, is a function of position and time. • General solution: l a, = y# a − + y (a + ) • Harmonic solutions: l a, = l sin a ± ~ • Wavelength, = 2/ and frequency y = ~/2. • Velocity, = ⁄y = ~⁄. The electromagnetic spectrum • In 1850, the only known forms electromagnetic waves were ultraviolet, visible and infrared. • The human eye is only sensitive to a very narrow range of wavelengths: Discovery of Radio Waves The Electromagnetic Spectrum Electromagnetic Waves nP Mo nP Mo = GH K H P np nLP • A solution is %c b, = % sin b − ~ where ~ = = 2/ • What is the magnetic field? 6m %c =− = −% cos b − ~ b 6m a, = % sin a − ~ ~ Electromagnetic Waves ax bz ly • %, 6 and are mutually perpendicular. • In general, the direction is ̂ = % × 6 Energy in Electromagnetic Waves Energy stored in electric and magnetic fields: 1 1 4 = % = = 6 2 27 For an electromagnetic wave, 6 = % ⁄ = % 7 1 1 = = 6 = % = 4 27 2 The total energy density is = = + 4 = % Intensity of Electromagnetic Waves • Intensity is defined as the average power transmitted per unit area. Intensity = Energy density × wave velocity % 8 = % = 7 7 = 377Ω ≡ (Impedance of free space) Poynting Vector • We can construct a vector from the intensity and the direction ̂ = % × 6: %×6 C = 7 % C = =8 • This represents the flow of power in the direction ̂ • Average electric field: %=2 = % / 2 % C = 2 • Units: Watts/m2 Have a nice spring break!